IRLF 


S  CIN  •'  /.  ]  i. 


IN  MEMORIAM 
FLOR1AN  CAJOR1 


THE   WESTERN^, . 

PRACTICAL  ARITHMETIC, 


WHEBE1N     THE 

RULES  ARE  ILLUSTRATE,  AND  THEIR  PRINCIPLES 
EXPLAINED: 

CONTAINING 

A  GREAT  VARIETY  OF  EXERCISES, 

PART^CULAJILY-ADAPTED  TO  THE 

CURRENCY  OF  THE  UNITED  STATES. 

WITH 

AN    APPENDIX:  ""S 

CONTAINING  THE  CANCELING  SYSTEM,  ABBREVIATIONS  IN  MULTIPLl-f 
CATION,  MENSURATION,  AND  THE  ROOTS.  -) 

DE«IQNEO  FOB  THE  USE  OF 

SCHOOLS  AND  PRIVATE  STUDENTa 

COMPILED 


BY    JOHN     L.    TALBOTT. 


PUBLISHED  BY  E.  MORGAN  &  CO., 

Ill    MAIN   STREET. 
1  853. 


District  of  Ohio,  to  wit :      ? 
District  Clerk's  Office.  5 

BE  IT  BEMEMBEBED,  that  on  the  seventeenth  day  of  January,  An> 
no  Domini  eighteen  hundred  and  forty-five,  E.  MOBGAN  &  Co.,  of 
the  said  District,  have  deposited  in  this  office  the  title  of  a  book,  the 
title  of  which  is  in  the  words  following,  to  wit : 

u  The  Western  Practical  Arithmetic,  wherein  the  rules  are  illus- 
trated, and  their  principles  explained,  containing  a  great  variety  of 
exercises,  particularly  adapted  to  the  currency  of  the  United  States : 
with  an  Appendix,  containing  the  Canceling  System,  Abbreviations 
in  Multiplication,  Mensuration  and  the  roots ;  designed  for  the  use 
of  schools  and  private  students;  compiled  by  JOHN  L.TALuoTT;"the 
right  whereof  they  claim  as  proprietors,  in  conformity  with  an  act  of 
Congress,  entitled  "An  act  to  amend  the  several  acts  respecting  copy- 
rights." WM.  MINER,  Clerk  of  District. 


PREFACE.         V&^T'f-' 

I.v  presenting  this  work  to  the  public,  the  author  makes  no  pre- 
tensions to  having  discovered  any  new  spring  by  which  to  put  the 
youthful  mind  into  action,  nor  any  new  method  of  communicating 
a  knowledge  of  Arithmetic.  He  has  founded  his  work  on  the  belief 
that  labor  and  labor  only,  can  insure  success  in  any  pursuit ;  and 
-that  labor  should  always  be  bestowed  upon  those  objects  which  pro- 
duce the  greatest  useful  result. 

In  the  selection  and  arrangement  of  matter,  therefore,  those  rules 
that  are  of  the  most  general  use,  have  been  presented  first,  and  their 
exercises  made  extensive,  that  the  pupil  many  early  become  familiar 
with  their  principles,  and  expert  in  their  application. 

The  explanations  accompanying  the  rules,  are  designed  to  facili- 
tate the  progress  of  private  students,  and  to  diminish  the  labor  of 
teachers,  especially  in  large  schools,  where  they  are  unable  to  give 
to  each  pupil  the  necessary  explanations. 

The  MESSUUATIOW  of  Carpenters',  Masons',  Plasterers'  and 
Pavers'  work,  &c.,  will  be  found  an  acceptable  part  of  Arithmetic, 
to  every  man  of  business,  and  a  practical  knowledge  of  it  will  con- 
tribute much  to  the  security  and  satisfaction  of  both  workmen  and 
employers,  in  estimating  amounts  of  work.  This  has  been  intro- 
duced in  consequence  of  numerous  applications  to  the  author  to 
measure  various  kinds  of  work,  and  for  instruction  in  particular 
rules  of  Mensuration. 

The  system  of  Book  Keeping,  is  thought  to  be  sufficient  for  all 
the  purposes  of  farmers,  mechanics  and  retailers,  in  that  necessary 
branch  of  a  business  education.  a*—  - 

How  far  the  author  has  succeeded  in  his  attempts  to  compile  a 
useful  work,  particularly  adapted  to  the  circumstances  of  the  Western 
People,  remains  for  them  to  judge,  and  for  experience  to  determine. 


NOTICE. 

THE  favorable  reception  of  this  treatise  and  the  increasing  demand 
for  it,  have  induced  the  publishers  to  revise,  enlarge,  and  otherwise 
improve  the  work.  Such  alterations  and  amendments  have  been 
made  as  the  experience  of  the  author  and  of  other  intelligent  and 
successful  teachers  has  suggested;  it  is  therefore  presumed,  that  the 
work  will  be  found  more  useful,  and  consequently  more  acceptable 
than  heretofore. 

Numerous  testimonials  to  the  merits  of  the  work,  have  been  re- 
ceived; but  its  general  adoption  without  any  efforts  to  force  its 
introduction,  and  its  intrinsic  worth,  are  our  main  reliance;  we  have 
therefore  given  it  a  thorough  revision,  and  now  submit  the  result  of 
our  labors  to  a  discerning  public. 

FebriiajyTJ841.  P  MORGAN  &  Co. 


_  -J.elk 

v5j 


CONTENTS. 

PACK. 

Numeration,  ••• ~ 5 

Simple  Addition, 12 

Simple  Subtraction, ••   15 

Simple  Multiplication, 18 

Simple  Division, • 23 

Addition  of  Federal  Money, - 30 

Subtraction  of  Federal  Money, 32 

Multiplication  of  Federal  Money, 34 

Division  of  Federal  Money, 36 

Reduction, 38 

Compound  Addition, •••    60 

Compound  Subtraction, • 66 

Compound  Multiplication,    72 

Compound  Division, 80 

Simple  Proportion, •••  •    88 

Compound  Proportion, 100 

Practice, 103 

Tare  and  Tret, 109 

Interest, 112 

Compound  Interest, 118 

Insurance,  Commission  and  Brokage, 121 

Discount,  122 

Equation  of  Payments, 124 

Barter, 126 

Loss  and  Gain, 128 

Fellowship, 132 

ar  Fractions, 135 

Reduction  of  Vulgar  Fractions,  • 136 

Addition  of  Vulgar  Fractions, 144 

Subtraction  of  Vulgar  Fractions, 146 

Multiplication  of  Vulgar  Fractions,. 147 

Division  of  Vulgar  Fractions, 148 

Decimal  Fractions, •. 149. 

Addition  of  Decimals, 150 

Subtraction  of  Decimals, ib. 

Multiplication  of  Decimals, 151 

Division  of  Decimals, -  • ib. 

Reduction  of  Decimals, • 153 

Proportion  in  Decimals, 155 

Compound  Proportion  in  Decimals, 156 

Mensuration, > • •••    ib. 

Involution, . .    167 

Evolution, 169 

Square  Root, ib. 

Cube  Root, 174 

Roots  of  All  Powers, 179 

Arithmetical  Progression 180 

Geometrical  Progression 184 

Appendix 194 

Exchange *.* 189 

Promiscuous  Exercise  .  .  ..••..  ,190 


ARITHMETIC. 


ARITHMETIC  is  that  part  of  MATHEMATICS  which 
treats  of  numbers.  It  is  both  a  science  and  an  art; — 
the  science  explains  the  nature  of  numbers,  and  the 
principles  upon  which  the  rules  are  founded^  while  the 
art  relates  merely  to  the  application  of  the  various 
rules. 

All  the  operations  of  arithmetic  are  conducted  by 
means  of  FIVE  fundamental  rates,  viz.,  Numeration, 
(which  includes  Notation,}  Addition,  Subtraction, 
Multiplication,  and  Division. 

NUMERATION  AND  NOTATION. 

Numeration  is  the  art  of  representing  figures  or  num- 
bers by  words ;  Notation  is  the  art  of  representing  num- 
bers by  characters  called  figures. 

'    All  numbers  are  represented  by  the  following  charac- 
ters, which  are  called  figures  or  digits. 

0,       1,      2,       3,      4,       5,      6,       7,       8,       0. 
nought,  one,  two,  three,  four,  five,  six,  seven,  eight,  nine. 

The  one  is  often  called  a  unit,  it  signifies  a  whole 
tb"«ig  of  a  kind ;  two  signifies  two  units  or  ones ;  three 
s  ^nifies  three  units  or  ones,  <fcc. 

The  value  which  the  figures  have  when  standing 
alone  is  called  their  simple  value ;  but  in  order  to  denote 
numbers  higher  than  9,  it  is  necessary  to  give  them  ano- 
ther value  called  a  local  value,  which  depends  entirely 
on  the  order  or  place  in  which  they  stand.  Thus,  when 
we  wish  to  write  the  number  ten  in  figures,  we  do  it 
by  combining  the  characters  already  known,  placing  a 
1  on  the  left  hand  of  the  0,  thus,  10,  which  is  read  ten. 
This  10  expresses  ten  of  the  units  denoted  by  1,  but 
as  it  is  only  a  single  ten  it  is  called  a  unit,  and  the 
1  being  written  in  the  second  order  or  second  place 
from  the  right  hand  to  express  it,  it  is  called  a  unit  of 
the  second  order,  the  first  place  being  called  the  place 


6  NUMERATION    AND    NOTATION. 

of  units,  and  the  second,  the  place  of  tens  ;  ten  units 
of  the  first  order  making  one  unit  of  the  second  order. 

When  units  simply  are  named,  units  of  the  first  order 
are  always  meant,  when  units  of  any  other  order  are 
intended,  the  name  of  the  order  is  always  added. 

Two  tens  or  twenty,  are  written       20. 

Three  tens  or  thirty,  "         "  30. 

Four  tens  or  forty,  "         "  40. 

Five  ten*  or  fifty,  "         "  50. 

Six  tens  or  sixty,  "         "  60. 

Seven  tens  or  seventy,  "         "  70. 

Eight  tens  or  eighty,  "         "  80. 

Nine  tens  or  ninety,  "         "  90. 

Ten  tens  or  one  hundred,     "         "          100. 

The  numbers  between  10  and  20,  between  20  and 
30,  between  30  and  40,  <fec.  may  easily  be  expressed 
by  considering  the  tens  and  units  of  which  they  are 
composed.  Thus,  eleven  being  composed  of  one  ten 
and  one  unit,  is  expressed  thus,  11,  twenty-thcee  being 
composed  of  two  tens,  and  three  units,  is  expressed 
thus,  23.  &c. 

Sixteen  being  1  ten  and  6  units,  is  written  thus,  16. 

Thirty-nine  being  3  tens  and  9  units,  is  written  39. 

Sixty-four  being  6  tens  and  4  units,  is  written  64. 

Ninety-five  being  9  tens  and  5  units,  is  written  95. 

Ten  tens  or  one  hundred  forms  a  unit  of  the  third 
order ;  it  is  expressed  by  placing  a  1  in  the  third  pin  e, 
and  filling  the  first  and  second  places  with  cyphei  . 
thus,  100.  Two  hundred  is  expressed  thus,  200 
Three  hundred  thus,  300,  <fcc. 

With  the  orders  of  units,  tens,  and  hundreds,  all  the 
numbers  between  one  and  one  thousand  may  be  readily 
expressed.  For  example,  in  the  number  four  hundred 
and  twenty-seven,  there  are  4  hundreds,  2  tens,  and  7 
units,  that  is,  4  units  of  the  third  order,  2  units  of  the 
second  order,  and  7  units  of  the  first  order. 

§    =   I 

Hence  the  number  is  written  thus,     427 

In  the  number  three  hundred  and  five,  there  are  3 
hundreds,  no  tens,  and  5  units,  or  3  units  of  the  third, 


NUMERATION    AND    NOTATION.  7 

none  of  the  second,  and  five  of  the  first  order,  hence  the 
number  is  written  thus,  £   *  I 

305 

Ten  units  of  the  order  of  hundreds,  that  is  ten  hun- 
dreds form  a  unit  of  the  fourth  order,  called  thousands, 
written  thus,  1000. 

In  the  same  manner  ten  units  of  the  fourth  order  form 
a  unit  of  the  fifth  order,  called  tens  of  thousands. 

The  following  may  be  regarded  as  the  principles  of 
Notation  and  Numeration. 

1st.  Ten  units  of  the  first  or  lowest  order,  make  one 
unit  of  the  second  order;  ten  units  of  the  second  order, 
make  one  unit  of  the  third  order,  and  universally  ten 
units  of  any  order  make  a  unit  of  the  next  higher 
order. 

2d.  Jill  numbers  are  expressed  by  the  nine  digits, 
and  the  cypher,  and  this  is  effected  by  giving  to  the 
same  figure  different  values  according  to  the  place  it 
occupies.  Thu^,  4  in  the  first  place  is  4  units,  in  the 
second  place  4  tens  or  forty,  and  so  on.  This  tenfold 
increase  of  value  by  changing  the  place  of  the  same 
figure  is  usually  expressed  by  saying  that  figures  in- 
crease from  right  to  left  in  a  tenfold  proportion.  The 
names  of  the  orders  are  to  be  learned  from  the 

NUMERATION   TABLE. 


3          2  o          « 

S§  •£        | 

**"  3  **""§• 

JS  aa  rS  "^  « 

|  |  |  |  ^  (| 

987654321 

The  orders  are  likewise  divided  into  periods  of  ei 
places  each,  according  to  the  following  table. 


NUMERATION    AND    NOTATION. 


of  Billions. 


ofMilli 


llioni. 


of  Unit*. 


f^ 

I 
ii 


II 


g    2 

' 


.  . 

s  o  'S  S  .3 

S3    £  X!    P    o»    £5 


0        "5 

W2   "5  "^        • 

!<=      - 


The  periods  succeeding  those  in  the  table,  are  Tril- 
lions, Quadrillions,  Quintillions,  Sextillions,  Sepiil- 
lions,  Octillions,  and  Nonnillions,  and  analogical  names 
might  be  formed  for  the  succeeding  higher  periods. 

From  the  preceding  remarks  the  pupil  will  readily 
understand  the  reason  of  the  following  rule  for  numer- 
ating or  expressing  figures  by  words. 

RULE. — Commence  at  the  right  hand,  and  separate 
the  given  number  into  periods,  then  beginning  at  the  left 
hand,  read  the  figures  of  each  period  as  if  they  stood 
alone,  and  then  add  the  name  of  the  period. 

Thus,  the  number  8304000508245,  when  divided 
into  periods,  becomes  8,304000,508245,  and  is  read, 
Eight  billion,  three  hundred  and  four  thousand  mil- 
lion, Jive  hundred  and  eight  thousand  two  hundred 
and  forty-Jive.  The  name  unit  of  the  right  hand  pe- 
riod is  commonly  omitted  in  reading. 

EXERCISES  IN  NUMERATION. 

Ex.  1.    35  10.  3700054 


2.  204 

3.  513 

4.  2000 

5.  3054 

6.  7428 
7  10345 
8.  40024 

X   9.  61304 


11.  6130425 

12.  2701030 

13.  3705423 

14.  6803217 

15.  2003005 

16.  70032004 

17.  62003005 

18.  91010010 


19. 
20. 
21. 
22. 
23. 
24. 
25. 
26. 
27. 


20031025 
68723145 
901023406 
820302008 
310275603 
600000501 
3000400230024 
80000102051003 
50000021375604 


28.  4000012000040250014 
29.  1000982000375000482000354000271000032561804 


NUMERATION    AND    NOTATION.  9 

From  the  preceding  tables  and  remarks,  the  pupil  will 
likewise  readily  understand  the  reason  of  the  following 
rule  for  notation,  or  expressing  numbers  by  figures. 

RULE. — Make  a  sufficient  number  of  cyphers  or  dots, 
and  divide  them  into  periods,  then  underneath  these 
dots  write  each  figure  in  its  proper  order  and  fill  the 
vacant  orders  with  cyphers. 

NOTE. — The  object  of  the  dots  or  cyphers,  being  to 
guide  the  learner  at  first,  after  a  little  practice  he  may 
dispense  with  them. 

Ex.  1.  Write  down  in  figures  the  number  twenty 
millions  three  hundred  and  four  thousand  and  forty. 
Here  millions  being  the  highest  period  named,  we 
write  cyphers  to  correspond  with  that,  and  the  period 
of  units,  and  then  underneath  these  place  the  significant 
figures  in  their  proper  order,  and  afterwards  fill  the 
vacant  orders  with  cyphers. 

000000,  000000 
20304040 

The  pupil  must  recollect  that  cyphers  being  of  no 
use  except  to  fill  vacant  orders,  are  never  to  be  placed 
to  the  left  of  whole  numbers. 


EXERCISES    IN    NOTATION. 

Express  the  following  numbers  in  figures. 

EXAMPLES. 

2.  Seventy-five. 

3.  Ninety. 

4.  •  One  hundred  and  five. 

5.  Three  hundred  and  twenty. 

6.  Nine  hundred  and  four. 

7.  Eight  hundred  and  ninety. 

8.  Two  thousand  three  hundred  and  five. 

9.  Six  thousand  and  forty. 

10.  Seven  thousand  and  four. 

11.  Eight  thousand  and  ninety-five. 

12.  Ten  thousand  five  hundred  and  fifty-six 

13.  Forty  thousand  and  forty. 

14.  Ninety-five  thousand  two  hundred  and  sixty-seven. 

15.  Eighty  thousand  one  hundred  and  nine. 


j]   10  NUMERATION    AND    NOTATION. 

•   16.  One  hundred  and  thirty-six  thousand  two  hundred 

and  seventy  five. 

)i  17.  Three  hundred  and  seven  thousand  and  sixty-four. 
Ji  18.  Five  hundred  thousand  and  five. 
||  19.  One  million,  two  hundred  and  forty-seven  thousand, 

four  hundred  and  twenty-three. 
|i  20.  Ten  millions,  forty  thousand  and  twenty. 

21.  Sixty  millions,  seventeen  thousand  and  two. 

22.  One  hundred  and  four  millions  two  hundred  and 

four  thousand  and  sixty- five. 

23.  Five  hundred  and  three  millions,  one  hundred  and 

two  thousand  and  nine. 

24.  Ninety  one  thousand  and  two  millions,  and  four. 

25.  Sixty  billions,  three  millions  and  forty-one  thousand. 

26.  One  billion,  one  hundred  million,  one  thousand  and 

one. 

The  Roman  method  of  representing  nnmbers,  is  by 
means  of  certain  capital  letters  of  the  Roman  alphabet. 
Thus: 


I 

II 

III 

IV 

V 

VI 

VII 

VIII 

IX 

X 

XI 

XII 

XIII 

XIV 

XV 

XVI 

XVII 


one 

two 

three 

four 

five 

six 

seven 

eight 

nine 

ten 

eleven 

twelve 

thirteen 

fourteen 

fifteen 

sixteen 

seventeen 


xvni 

XIX 

XX 

XXX 

XL 

L 

LX 

LXX 

LXXX 

xc 
c 

cc 

ccc 

cccc 

D 

M 

MDCCCXXXVIII 


eighteen 

nineteen 

twenty 

thirty 

forty 

fifty 

sixty 

seventy 

eighty 

ninety 

one  hundred 

two  hundred 

three  hundred 

four  hundred 

five  hundred 

one  thousand 

1838 


NOTE  1 .  As  often  ag  any  letter  is  repeated,  so  often  is  its  vilue  re- 
peated. 

NOTE  2.  A  less  character  before  a  greater  one,  diminishes  its  value 
NOTE  3.  A  les*  character  after  a  greater  one,  increases  its  value. 


EXPLANATION    OF    CHARACTERS.  11 

QUESTIONS. 

What  is  Arithmetic  ?  When  is  it  a  science  ?  When 
is  it  an  art  ?  What  are  the  fundamental  rules  of  arith- 
metic  ?  What  is  numeration  ?  What  is  notation  ?  What 
does  a  unit  signify  ?  What  does  two  signify  ?  Three, 
<fec.  ?  What  is  meant  by  the  simple  value  of  a  unit? 
What  does  the  local  value  of  a  figure  depead  on  ?  How 
do  you  write  the  number  ten  in  figures  ?  Why  is  the 
one  in  this  case  called  a  unit  of  the  second  order?  How 
many  units  of  the  first  order  does  it  take  to  make  a  unit 
of  the  second  order  ?  How  many  units  of  the  second 
order  does  it  require  to  form  a  unit  of  the  third  order? 
&c.  Repeat  the  principles  of  notation  and  numeration. 
l|  Repeat  the  names  of  each  of  the  first  nine  orders  as  ex- 
[i  pressed  in  the  numeration  table.  Repeat  the  name  of 
each  of  the  periods.  Repeat  the  Rule  for  numeration. 
Repeat  the^  Rule  for  notation. 


EXPLANATION  OF  CHARACTERS. 

Signs.  Significations. 

=  equal;  as  20s.  =  £  1. 

-f-  more ;  as  6  -f-  2  =  8.  ^ 

—  less  ;  as  8  —  2  =  6. 

X  into,  with,  or  multiplied  by  ;  as  6  X  2  =  12. 
-J-  by  (i.  e.  divided  by  ;)  as  6  -f-  2  =  3 ;  or,  2)6(3. 
:  : :  :      proportionality;  as  2  :  4  : :  6  :  12. 
J  or,  J  Square  Root ;  as  J  64  =  8. 
J             Cube  Root;  as  J  64  =  4. 
iy              Fourth  Root ;  as  J  16  =  2,  &c. 
A  vinculum ;  denoting  the   several  quantities 

over  which  it  is  drawn,  to  be  considered  jointly 

as  a  simple  quantity. 


12  SIMPLE    ADDITION. 

SIMPLE  ADDITION. 

SIMPLE  ADDITION  is  the  art  of  collecting  several  nuin- 
jers,  of  the  same  name,  into  one  sum. 

RULE. 

Place  the  numbers  with  units  under  units,  tens  under 
tens,  &,c.  Begin  the  addition  at  the  units,  or  right  hand 
column,  and  add  together  all  the  figures  in  that  column ; 
then,  if  the  amount  be  less  than  ten,  set  down  the  whole 
sum:  but  if  greater  than  ten,  see  how  many  tens  there 
are,  and  set  down  the  number  above  the  even  tens,  and 
carry  one  for  each  ten  to  the  next  column,  and  proceed 
with  it  as  in  the  first. 

Proof. — Begin  the  addition  at  the  top  of  each  column, 
and  proceed  as  before,  and  if  the  result  be  the  same,  it 
is  presumed  to  be  right, 

EXAMPLES. 

(1)  (2)  (3)  (4) 

432  231  214  4 

213  413  121  2 

121  121  312  5 

213  132  321  3 


979  sum        897  sum        968  sum     1  4  sum 


(5)  Here  4, 2, 1,  Sand  6  make  15.     In  fifteen  there 

27636  *s  one  ^en  aiu'  ^ve  lln'ls*  Set  down  *he  f've  units 
7  Q  ft  Q  2  un^er  tne  umts  c°luinn*  and  carry  one  for  the  ten 
'  *  to  the  next  or  tens  column. 

38941 

67832  Then  1,4,  3, 4, 9  and  3  make  24;  in  24  there  are 
59244  two  tens,  and  four  over:  set  down  the  foui  under 
the  column  of  tens,  and  carry  two  to  the  next  or 
hundreds  column  &c.,  to  the  last,  where  thr  whole 


273545      amount  may  be  set  down. 


SIMPLE   ADDITION.                                          13 

(6) 

(7) 

(8) 

47386 

99786 

72752 

29492 

86937 

37823 

18583 

27849 

78794 

89294 

49878 

23567 

28887 

72937 

98372 

74392 

48732 

12345 

288034 

386119 

323653 

(9) 

(10) 

(11) 

47823 

72683 

84736 

73714 

95892 

78928 

27834 

82783 

27849 

23925 

94973 

63782 

67883 

76892 

28637 

62734 

43987 

73862 

(12) 

(13) 

(14) 

73684 

9376 

7379 

7  5 

723 

7463 

473 

8  7 

729 

6893 

9 

489 

7 

4  8 

7  2 

483 

937 

6  8 

9  6 

9  2 

432 

APPLICATION. 

1.  Add  224  dollars,  365  dollars, 

427  dollars,  and  784 

dollars,  together. 

224 

365 

427 

784 

Answer^ 

$1800  Dollars 

14  SIMPLE   ADDITION. 

2.  Add  3742  bushels,  493  bushels,  927  bushels,  643 
bushels,  and  953  bushels,  together. 

Answer,  6758  bushels. 

3.  Add   7346  acres,  9387  acres,   8756  acres,  8394 
acres,  32724  acres.  Ans.  66607  acres. 

4.  Henry  received  at  one  time  15  apples,  at  another 
115,  at  another  19.     How  many  did  he  receive? 

Ans.  149. 

5.  A  person  raised  in  one  year  724  bushels  of  corn, 
in  another  3498  bushels,  in  another  9872.     How  much 
in  all?  Ans.  14094  bushels. 

6.  A  rnan  on  a  journey,  travelled  the  first  day  37 
miles,  the  second  33  miles,  the  third  40  miles,  the  fourth 
35  miles.     How  far  did  he  travel  ill  the  four  days? 

Ans.  145  miles. 

7.  A  has  a  flock  of  sheep  containing   34.     B  has  a 
flock  of  47,  and  C  of  fifty-four.     How  many  sheep  are 
there  in  the  three  flocks?  Ans.  135. 

8.  The  distance  from  Philadelphia  to  Bristol  is  20 
miles;  from  Bristol  to  Trenton,  10  miles;  from  Trenton 
to  Princeton,  12  miles;  from  Princeton  to  Brunswick, 
18  miles ;  from  Brunswick  to  New  York,  30  miles.    How 
many  miles  from  Philadelphia  to  New  York?    Ans.  90. 

9.  A  person  bought  of  one  merchant,  10  barrels  of 
flour,  of  another  20  barrels,   of  another  95    barrels. 
How  many  barrels  did  he  buy  ?         Ans.  125  barrels. 

10.  A  wine-merchant  has  in  one  cask  75  gallons,  in 
another  65,  in  a  third  57,   in  a  fourth  83 ;  in  a  fifth  74, 
and  in  a  sixth  67  gallons.     How  many  gallons  has  he 
in  all  f  Ans.  421  gallons. 

Questions. 

How  many  primary  rules  of  Arithmetic  are  there  f 

What  are  they  called  ? 

What  is  addition? 

How  do  you  place  numbers  to  be  added? 

Where  do  you  begin  the  addition  »* 

Why  do  you  carry  one  for  ten,  in  preference  to  any 
other  number? 

Ans.  Because  it  takes  ten  ones  to  make  one  ten,  ten 
tens  to  make  one  hundred,  &c.  (See  table,  page  9.) 


SDIPLE    SUBTRACTION.  15  | 

SIMPLE  SUBTRACTION. 

SIMPLE  SUBTRACTION  is  taking  a  less  number  from  a 
greater,  of  the  same  name,  to  show  the  difference  be- 
tween them. 

The  greater  number  is  called  the  minuend. 

The  less  number  is  called  the  subtrahend. 

The  difference,  or  what  is  left,  is  called  the  remainder 


RULE. 

Place  the  less  number  under  the  greater,  with  units 
under  units,  tens  under  tens,  &LC. 

Then  draw  a  line  under  them;  begin  at  the  right 
hand  or  units  place,  and  subtract  each  figure  of  the  sub- 
trahend from  the  figure  of  the  minuend  that  is  above 
it,  and  set  the  remainder  below.  When  the  figure  in 
the  subtrahend  is  greater  than  the  one  above  it,  borrow 
one  (which  is  one  ten)  from  the  next  figure,  and  add  it 
to  the  figure  of  the  minuend;  then  subtract  from  the 
sum. 

Proof. — Add  the  remainder  and  the  subtrahend  to- 
gether, and  if  the  sum  equal  the  minuend,  the  work  is 
!  presumed  to  be  right. 

EXAMPLES. 

(1)  (2) 

79252743  Minuend  9738476 

34120312  Subtrahend  2614253 


451324!'  1  Remainder  7124223 


(3)  Here  we  cannot  take  seven  from  two ;  then  we 

726398   -j  must  borrow  one  from  the  8:  that  one  is  one  fen; 
R  A  9    Pi   r    Q   >y  tnen  len  an<^  two  are  *vvelve  >  now  ta^e  seven  from 
'  twelve,  and  five  remain. 

One  is  borrowed  from  the  8,  leaving  only  7 ; 

838345  then  take  3  from  7,  and  4  remain  :  or,  suppose  8 

to  remain  untliminished ;  and  to  cancel  the  one  which  is  borrowed  from 

the  8,  add  one  to  the  3  below,  making  four;  then  four  from  eight  and 

four  remais-j  as  before,  &c. 


16  SIMPLE   SUBTRACTION. 

(4)  (5) 

9273847        82703682 
2641386        27341237 


6632461        55362445 


(6)  (7) 

7837286      273683070 
3273195          4321725 


4564091       269361345 


(8)  W 

68427362       593784283 
34613524        54321432 


(10)  (11) 

792836842        92037842 
24653128        41372761 


APPLICATION. 

1.  From  78  take  32  and  what  will  remain? 

Answer,  46. 

2.  From  478  take  324.     What  will  remain? 

Ans.  154. 

3.  Charles  had  723  apples,  and  sold  421.     How  ma- 
ny has  he  left?  Ans.  302 

4.  James  had  9768  dollars,  and  gave  for  a  house  and 
lot  3453  dollars.  How  many  has  he  left?     Ans.  6315, 

5.  A  farmer  had  3849  acres  of  land;  he  gave  to  his 
sons  2135  acres.     How   many   acres   has   he  left   for 
himself?  Ans.  1714. 

6.  There  are  two  piles  of  bricks,  one  contains  7SUH, 
and  the  other  4389.     How  many  more  are  there  in  the 
oae  than  in  the  other?  Ans.  3507. 


SIMPLE   SUBTRACTION.  17 

7.  Bought  100  bags  of  coffee,  weighing  14510  Ibs., 
and  sold  thereof  63  bags  weighing  6871  pounds;  how 
many  bags,  and  how  many  pounds  remain  unsold? 

Ans.  37  bags,  and  7639  Ibs. 

8.  A  man  bought  a  chaise  for  175  dollars,  and  to  pay 
for  it  gave  a  wagon  worth  37  dollars,  and  the  rest  in 
money.    How  much  money  did  he  pay  ? 

Ans.  138  dollars. 

9.  A  man  deposited  in  bank  8752  dollars,  and  drew 
out  at  one  time  4234  dollars,  at  another  1700  dollars, 
at  another  962  dollars,  and  at  another  49  dollars.     How 
much  had  he  remaining  in  bank?     Ans.  1807  dollars. 

10.  A  merchant  bought  4875   bushels  of  wheat,  and 
sold  2976  bushels.     How  many  bushels  remain  in  his 
possession?  Ans.  1899. 

1 1.  A  grocer  bought  25  hogsheads  of  sugar,  containing 
250  hundred  weight,  and  sold  9  hogsheads,  containing 
75  hundred  weight.     How  many  hogsheads  and  how 
many  hundred  weight  had  he  left  ? 

Ans.  16  hogsheads,  and  175  hundred  weight. 

12.  A  traveller  who  was  1300  miles  from  home,  trav- 
elled homeward  235  miles  in  one  week;  in  the  next  275 
miles;  in  the  next  325  miles;  and  in  the  next  290  miles. 
How  far  had  he  still  to  go,  before  he  would  reach  home  ? 

Ans.  175  miles. 

Questions. 

What  is  subtraction? 

What  is  the  greater  number  called  ? 

What  is  the  less  number  called? 

What  is  the  difference  called? 

How  do  you  place  numbers  for  subtraction? 

Where  do  you  begin  the  subtraction? 

When  the  lower  figure  is  greater  than  the  upper  one, 
how  do  you  proceed? 

Why  is  the  one  you  borrow,  one  ten. 

Ans.  Because  ten  ones  make  one  ten;  and  if  I  borrow 
one  ten  it  will  make  ten  ones  again,  &c. 

How  do  you  prove  subtraction' 

M^MMMMHWBMMMMMHMMHMMMMMBIMMHB^^ 

Vi* 


18                               SIMPLE 

MULTIPLICATION. 

SIMPLE  MULTIPLICATION. 

I     SIMPLE  MULTIPLICATION  is  a  short  method  of  perform- 

ing particular  cases  of  addition. 

The   number  to  be  multiplied,  is  the  multiplicand. 

The   number  to 

be 

multiplied 

by, 

is  the  multiplier. 

The  number  produced  is  the  product. 

The  multiplicand  and  multiplier  are  sometimes  called 

factors. 

MULTIPLICATION 

TABLE. 

Twice 

3  times 

4  times 

5  times 

6  times 

7  times 

1  make  2 

1  make  3 

1  make  4 

1  make  5 

1  make  6 

1  make  7 

2            4 

2            6 

2 

8 

2 

10 

2 

12 

2           14 

3            6 

3            9 

3 

12 

3 

15 

3 

18 

3          21 

4            6 

4          12 

4 

16 

4 

20 

4 

24 

4          28 

5          10 

5          15 

5 

20 

5 

25 

5 

30 

5          35 

6          12 

6          18 

6 

24 

6 

30 

e 

36 

6          42 

7          14 

7          21 

7 

28 

7 

35 

7 

42 

7          49 

8          16 

8          24 

8 

32 

8 

40 

8 

48 

8          56 

9          18 

9          27 

9 

36 

q 

45 

9 

54 

9          63 

10          20 

10          30 

10 

40 

10 

50 

10 

60 

10          70 

11          22 

LI          33 

11 

44 

11 

55 

11 

66 

11          77 

12          24 

12          36 

12 

•      48 

J2 

60 

12 

72 

12          84 

8  times           9  times 

10  times 

11  times 

12   times 

1  make    £ 

\     1  make 

9 

1  make  10 

1   make  11 

1    make  12 

2            1( 

5     2 

18 

2 

20 

2 

22 

2             24 

3            & 

[     3 

27 

3 

30 

3 

33 

3             36 

4            3S 

}     4 

36 

4 

40 

4 

44 

4            48 

5            4( 

5 

45 

5 

50 

5 

55 

5            60 

6        '    46 

}     6 

54 

G 

60 

6 

66 

6            72 

7            5( 

5     7 

63 

7 

70 

7 

77 

7            84  i 

8            W 

[     8 

72 

8 

80 

8 

88 

[ 

3            96 

9             7$ 

I     9 

81 

9 

90 

9 

99 

9           108 

10            8( 

1  10 

90 

10 

100 

10 

110 

10           120 

11             8* 

j  11 

99 

11 

110 

11 

121 

11            132 

12             9( 

i  12 

108 

12 

120 

12 

132 

12           144 

CASE  1. 

When  the  multiplier  does 

not 

exceed 

12. 

RULE.  —  Place  the  multiplier  under 

the  units 

figure  of 

he  multiplicand;  and 

multiply  each  figure  of  the  multi- 

I'licand  in  succession, 

and 

set  down  the  amount,  and 

:arry,  as  in  addition. 

Proof.  —  Multiply  the  multiplier  by 

the  multiplicand. 

SOIFLE   MULTIPLICATION.  19 
EXAMPLES. 

4231  Multiplicand          34253  7342 

2  Multiplier                              3  4 


8462  Product      102759    29368 


36563    8375    4378     9286 
567          8 


182815  50250  30646   74288 


4375    7862    3724     7482 
9        10        11        12 


39375   78620   40964   897^4 


EXERCISES 

1  Multiply          4218  by    2  Product  8436 

2  7321  by     3  2196  I 

3  87692  by    4  3507t>e» 

4  900078  by     9  8100702 

5  826870  by  10  8268700 

6  278976  by  11  3068736 

7  569769  by  12  6837228 

CASE  2. 
When  the  multiplier  exceeds  12. 

RULE. — Place  the  multiplier  as  before,  with  units  un- 
der units,  &c.  Then  multiply  all  the  figures  of  the  mul- 
tiplicand by  the  units  figure  of  the  multiplier,  setting 
down  the  product  as  before. 

Proceed  with  the  tens  figure  in  the  same  manner,  ob- 
serving to  set  the  product  of  the  first  figure  in  the  tens 
place  and  with  the  hundreds  figure  placing  the  first 
product  in  hundreds  place,  &.C.,  and  add  the  several  pro 
ducts  together. 


20 


SIMPLE   MULTIPLICATION. 
EXAMPLES. 


43752  Here  we  multiply  by  the  6  or  units  figure 

436  as  before:  then  by  the  3  or  tens  figure,  pla- 

cing  the  first  product  in  the  second  or  tens 

262512  place,  immediately  under  the  three,  in  the 

131256  multiplier.     In   like  manner  we  use  the  4, 

175008  placing  the  first  product  in  the  third  or  Atm- 

dreds  place,  immediately  under  the  4;  alter 

19075872  which  we  add  the  several  products  together, 

and  the  work  is  done. 


73684 
427 

515788 
147368 
294736 

31463068 


1  Mu  tiply 

2 

3 

4 

5 

6 

7 

8 

9 
10 
11 
12 
13 
14 
15 
16  > 
17 
18 


EXEBCISES. 

4736  by 

5762  by 

6483  by 

7368  by 

4327  ,y 

7382  by 

4728  by 

7584  by 

5678  by 

7G83  by 

4962  by 

7384  by 

4376  by 

7923  by 

6842  by 

7648  by 

8473  by 

9372  by 


37462 
563 


112386 
224772 
187310 

21091106 


34  Product  161024 


43 
54 
45 
56 
67 
76 
87 
78 
89 
98 
87 
97 
78 
89 
523 
456 
567 


247766 
350082 
331560 
242312 
494594 
359328 
659808 
442884 
683787 
486276 
642408 
424472 
617994 
608938 
3999904  !| 
3863688  1 
5313924  | 


SIMPLE  MULTIPLICATION.  21 

NOTE  1. — When  either  or  both  of  the  factors  have 

noughts  on  the  right  hand,  they  may  be  omitted  in  the 

operation,  and  annexed  to  the  product.     Thus : 

47  I  000  734  I  00 

42     00  42     000 


94  1468 

188  2936 


-Product  197400000  Product  3082800000 

NOTE  2. — When  the  multiplier  is  the  exact  product 
of  any  two  factors  in  the  multiplication  table,  the  opera- 
tion may  be  performed  by  separating  th«  multiplier  into 
its  components,  and  multiplying  first  by  the  one,  then 
its  product  by  the  other.  Thus : 

754  by  36         754  754  754 

9  3  6  36 

2272  4524  4524 

12  6  2262 


27144  27144  27144  27144 

9  and  4,  or  3  and  12,  or  6  and  6,  multiplied  together, 
produce  36,  and  by  using  either  pair,  according  to  the 
above  note,  the  true  result  is  obtained. 

EXERCISES. 

1  Multiply          756     by     42  Product  31752 

2  645  by     24  15480 

3  876  by     48  42048 

4  963  by     56  53028 

5  827  by     72  59544 

6  946  by     81  76626 

7  875  by     84  73500 

8  948  by     96  91008 

9  795  by  108  85860 
Tiie  pupil  may  work  these  by  all  the  several  pairs  of 

components  that  he  can  find  in  the  multiplier. 


22 


SIMPLE  MULTIPLICATION. 


NOTE  3. — When  the  multiplier  is  not  the  exact  pro 
duct  of  any  two  numbers  in  the  table,  use  two  factors 
whose  product  is  short  of  the  multiplier,  then  multiply 
the  sum  by  the  number  required  to  supply  the  deficiency 
and  add  its  product  to  that  obtained  by  the  two  .factors. 


583X3 
4 


11680 
1749 

13409 

1  Multiply 

2 

3 

4 

5 


EXAMPLES. 

583X2 

7 


12243 
1166 


583X5 
3 

1749 
6 

10494 
2915 


13409 


13409 


13409 


EXERCISES. 


846  by  26 

784  by  29 

975  by  34 

859  by  43 

794  by  59 


Product  21996 
22736 
33150 
36937 
46846 


PROMISCUOUS  EXERCISES. 


1.  Charles  has  24  marbles,  and  John  has  13  times  as 
many;  how  many  has  John?     -  Ans.  312. 

2.  A  gentleman  owns  17  houses,  for  each  of  which  he 
receives  250  dollars  rent;  how  much  does  he  receive 
for  them  all?  Ans.  4250. 

3.  A  laborer  hired  himself  to  a  farmer  for  1 1  years, 
at  150  dollars  a  year;  how  much  did  he  receive? 

Ans.  1650  dollars. 

4.  A  person  wishes  to  purchase  26  shares  of  Bank 
stock  at  75  dollars  a  share ;  what  must  he  pay  ? 

Ans.  1950  dollars. 

5.  A  mason  having  built  a  house,  found  that  98470 
bricks  were  in  it;  suppose  he  desires  to  build  19  such 
houses,  how  many  bricks  must  he  obtain  for  the  pur- 


pose 


Ans.  1870930. 


SIMPLE    DIVISION.  23 

SIMPLE  DIVISION. 

SIMPLE  DIVISION  is  a  short  method  of  performing  sev- 
eral subtractions. 

The  number  to  be  divided  is  called  the  dividend. 

The  number  by  which  it  is  to  be  divided  is  called  the 
divisor. 

The  number  of  times  that  the  divisor  is  contained  in 
the  dividend,  is  called  the  quotient. 

So  many  figures  of  the  dividend  as  are  taken  to  be 
divided  at  one  time,  is  called  a  dividual. 

If  any  thing  remain  when  the  operation  is  completed, 
it  is  called  the  remainder. 


CASE  i. — SHORT  DIVISION. 
When  the  divisor  does  not  exceed  12. 

RULE. — Place  the  divisor  on  the  left  hand  side  of  the 
number  to  be  divided. 

Consider  how  often  the  divisor  is  contained  in  the  first 
figure  or  figures  of  the  dividend,  and  set  down  the  result 
below ;  observing  how  many  remain,  if  any.  If  there 
be  no  remainder,  consider  how  often  the  divisor  is  con- 
tained in  the  next  figure :  but  if  there  be  a  remainder, 
call  it  so  many  tens,  and  add  the  next  figure  to  it,  and 
divide  the  sum,  placing  the  result  beneath,  as  before. 

Proof. — Multiply  the  quotient  by  the  divisor;  add  the 
remainder,  if  any,  and  the  product  will  equal  the  divi- 
dend. 

EXAMPLES. 

Dividend. 
Divisor  2)182        2)648  3)963  4)484 

Quotient   241  324 


24  SIMPLE   DIVISION. 

3)741851  Here  3  are  contained  In 

7  two  times*  and  one  re- 

o/iTOQO  I  Q  T*^n     \*A^   mains ;  place  the  two  under 

247283-H  Remainder  theseven,  and  suppose  the 

one  that  remains  to  be  one 

ten,  and  add  the  next  fig- 

Proof    741851  ure  (4)  to  »*>  which  make3 

fourteen. 

Now  3  are  contained  in  fourteen  4  times,  and  2  remain.  Set  the 
4  down  under  the  4  in  the  dividend,  and  suppose  the  two  that  remain 
to  be  two  fen$,  and  add  the  next  figure  (1)  to  it,  which  make  twenty- 
one.  Now  3  into  21  go  7  times,  and  no  remainder.  Place  the  7  un- 
der the  1  in  the  dividend,  and  proceed  in  the  same  manner  with  the 
other  figures. 


4)65270167  5)6572686 

163175414-3  Rem.  1314537+1  Rem 

4  5 


Proof       C5270167  Proof    6572G86 

6)8739627  7)4873692 

8)9273684  0)8379286 

10)946873  11)893726  12)98796 


SIMPLE    DIVISION.  25 

EXERCISES. 

Divide     7893762  by    6  Ans.  1315627 

9387984  by    7  ,      1341140—4 

6928437  by    8  866054—5 

9276874  by     9  1030763—7 

8672934  by  10  867393—4 

6873842  by  11  624894—8 

7369287  by  12  614107—3 

CASE  2. — LONG  DIVISION. 
When  the  divisor  exceeds  12. 

RULE. — Place  the  divisor  to  the  left  hand  of  the  divi- 
dend, as  in  case  1. 

Consider  how  often  the  divisor  is  contained  in  the 
least  number  of  figures  into  which  it  can  be  divided; 
and  set  down  the  result  at  the  right  hand  of  the  dividend. 

Multiply  the  divisor  by  the  quotient  figure  thus 
found,  and  set  the  product  under  the  dividual  or  figures 
supposed  to  be  divided. 

Subtract  the  product  from  the  dividual,  and  set  down 
what  remains.  Bring  down  the  next  figure  of  the  divi- 
dend, and  proceed  as  before,  till  all  the  figures  are 
brought  down  and  divided. 

EXAMPLES. 
Divisor.  Dividend.  Quotient. 

27)984376(36458  Twenty-seven  into  98  go  3 

81  times:    multiply  the  divisor 

(27)  by  3  and  set  the  product 
under  the  dividual  (98)  and 
subtract.  To  the  remainder 
(17)  bring  down  the  next  fig- 
ure (4)  of  the  dividend.  Now 
27  into  174  go  6  times.  Place 
the  6  in  the  quotient  and  mul- 
tiply (27)  the  divisor,  by  6, 
and  set  the  product  under  174 

and  subtract  as  before,  &,c. 

226 
216 
10  Remainder. 


26                                         SIMPLE   DIVISION. 

Divisor.  Dividend.   Quotient. 
42)    98754   (2351 
84              42  Divisor 

147         4702 
126       9404 
12  Remainder 
215     
210    98754  Proof 

54 
42 

12  Remainder 

32)789627(24675               65)1827538(28115 
64                                        130 

149                                         527 
128                                         520 

216 
192 

242 
224 

187 

75 
65 

103 
65 

388 

160 

325 

27  Rem. 

63  Rem. 

EXEltCIgES. 

Divide     8769 

by 

13 

Quo.  674 

Rem.  7 

476 

by 

15 

31 

11 

958 

by 

18 

53 

4 

1475 

by 

28 

52 

19 

4277 

by 

31 

137 

30 

25757 

by 

37 

696 

5 

1 

63125 

by 

123 

513 

26 

1 

253622 

by 

422 

601 

SIMPLE    DIVISION. 


27 


NOTE  1. — Cyphers  on  the  rigkt^hand  of  the  divisor 
may  be  omitted  in  the  operation,  observing  to  separate 
as  many  figures  from  the  right  of  the  cttvidend,  which 
must  be  annexed  to  the  remainder. 


EXA3TPLES. 


54 


00)1463 
108 

383 
378 

Rem.  540 


40(27 


32  |  0)7617 
64 

121 
96 

257 
256 


3(238 


Rem.  13 

EXERCISES.  Jf| 

Divide     40220  by     1900  Ans.  21  Keai.  320 

137000  by     1GOO            85  1000 

99607765  by  27000        3689  4765 

2304108  by    5800          397  1508 

NOTE  2. — When  the  divisor  is  the  exact  product  of 
any  two  numbers  in  the  multiplication  table,  the  opera- 
tion may  be  performed  by  dividing  first  by  one  of  the 
component  parts,  and  then  the  quotient  by  the  other. 

To  get  the  true  remainder,  multiply  the  last  remain- 
der by  the  first  divisor,  and  add  the  first  remainder. 


EXAMPLES. 


7     98754 


42 


[o  |  14107  —  5  first  remainder 

2351  —  1  last  remainder 
7  first  divisor 


add 


|i 


5  first  remainder 
12  true  remainder 


28  SIMPLE   DIVISION. 

984376 
27   * 

'  328125—1 


36458~-3  X  3  +  1  =10  Rem 

EXERCISES. 

9756  by  35  Quotient  278  Rcm.  26 

8491  by  81  104            67 

44767  by  18  2487               1 

92017  by  56  1643              9 

38751  by  48  807             15 

7S4071  by  72  10195            31 

APPLICATION. 

1.  SevWn  boys  have   161  apples,  which  they  divide 
equally  among  them.     How  many  does  each  have? 

Answer,  23. 

2.  What  is  the  quotient,  if  8736  be  divided  by  8,  and 
that  quotient  by  4?  Ans.  273. 

3.  If  350  dollars  be  equally  divided  among  7  men, 
what  will  be  the  share  of  each  ?  Ans.  50. 

4.  How  many  times  are  27  contained  in  952? 

Ans.  35  times  and  7  over. 

5.  Suppose  2072  trees  planted  in  14  rows.    How  ma- 
ny trees  will  there  be  in  each  row?  Ans.  148. 

6.  Several  boys  who  went  to  gather  nuts,  collected 
4741,  of  which  each  boy  received  431.     How  many 
boys  were  there?  Ans.  11. 

7.  If  the  expense  of  erecting  a  bridge,  which  is  15036 
dollars,  be  equally  defrayed  by  179  persons,  what  must 
each  pay?  Ans.  84  dollars. 

8.  Suppose  a  man  receive  in  one  year  2920  dollars; 
how  much  a  day  is  his  income  at  that  rate;  and  sup- 
pose that  his  expenses  for  the  year  amount  to  1769  dol- 
lars.    How  much  will  he  save  in  a  year? 

Ans.  His  income  will  be  8  dollars  a  day;  he  will  save 
1151  dollars  in  a  year. 


SIMPLE    DIVISION.  29 

Questions. 

What  is  division? 

What  do  you  call  the  number  that  is  to  be  divided? 

What  do  you  call  the  number  you  divide  by  ? 

What  do  you  call  the  number  obtained  by  division? 

What  do  you  call  that  which  is  left  when  the  work  is 
done  ? 

When  the  divisor  does  not  exceed  12,  how  do  you  per- 
form the  operation? 

When  the  divisor  exceed  12,  how  do  you  proceed? 

How  do  you  prove  division? 

How  may  the  operation  be  performed  when  there  are 
cyphers  at  the  right  hand  of  the  divisor? 

How  may  it  be  performed  when  the  divisor  is  the  exact 
product  of  two  numbers  in  the  multiplication  table? 

How  do  you  obtain  the  true  remainder  in  the  last  case? 


PROMISCUOUS    EXERCISES   iN   THE   PRECEDING   RULES. 

1.  If  the  contents  of  five  bags  of  dollars,  containing 
$295,    $410,    $371,    $355,    and   $520,    be    divided 
equally   among  25  persons,  how  much  is  the  share  of 
each?  Ans.  $78.04 

2.  A  man  possessed  of  an  esnue  of  $30,000,  disposed 
of  it  in  the  following  manner:  to  his  brother  he  gave 
$1500,  and  the  balance  to  his  5  sons,  to  be  equally  di- 
vided among  them.     What  was  each  one's  share  ? 

Ans.  $5700. 

3.  What  number  is  it,  which  being  added  to  9709 
will  make  110901?  Ans.  101192. 

4.  Add  up  twice  3?7,  three  times  794,  four  times 
31196,  five  times   15S80,  six  times  95280,  and   once 
33304.  Ans.    812,344. 

5.  Three  merchants  have  a  stock  of  14876  dollars, 
of  which  A  owns  4963  dollars,  B  5188,  and  C  the  re- 
mainder.    How  much  does  C  own?    Ans.  4725  dolls. 


30  FEDERAL   MONEY. 

FEDERAL  MONEY, 

OR  MONEY  OF  THE  UNITED  STATES. 

TABLE. 

10  mills  make  1  cent 

10  cents  1  dime 

10  dimes  1  dollar 

10  dollars  1  eagle 

These  denominations  bear  the  same  relation  to  each 
other  as  those  of  units,  tens,  hundreds,  &c.  Federal 
money  is  therefore  added,  subtracted,  multiplied,  and 
divided  by  the  same  rules  as  Simple  Addition,  Subtrac- 
tion, Multiplication,  and  Division. 

ADDITION  OP  FEDERAL  MONEY. 

Rule. 

Place  the  numbers  one  under  another,  with  mills  on 
the  right,  cents,  dimes,  &,c.,  in  succession ;  observing  to 
keep  mills  under  mills,  cents  under  cents,  &c.  Then 
proceed  as  in  simple  addition. 

When  halves  or- fourths  of  a  cent  occur,  find  their 
amount  in  fourths,  and  consider  how  many  cents  these 
fourths  will  make,  and  carry  them  to  the  column  of  cents. 

EXAMPLES. 

Eagles.  Dolls.  Dimes.  Cents  Mills.    Dolls.  Ds.  Cts. 

789 
978 
684 
637 
482 

E.   

27        3          6          453570 

NOTE. — In   common  business    transactions,    eagles, 
dimes,  and  mills  are  not  used:  dollars,  cents,  and  frac- 
tions of  a   cent,  are  the  only  denominations  kept  in 
j  accounts. 


3 

7 

8 

9 

5 

7 

4 

9 

8 

7 

2 

3 

8 

7 

9 

8 

8 

9 

8 

6 

4 

7 

8 

9 

8 

FEDERAL  MONEY. 


EXAMPLES. 


Ds.  cts. 
34   62 


56 
27 
23 
27 


31 

82 
68 
42 


169  ,  85 

Ds.  cts. 
468  ,  31 
723  ,  62 

845  ,  92 
736  ,  25 

846  ,  31 
428  ,  62 


EXERCISES. 

(2) 
Ds.  cts. 

927  ,  24 
768  ,  32 
427  ,  56 
792  ,  34 

587  ,  62 
842  ,  27 


Ds.  cts. 
427  ,  68 
342  ,  31 
427  ,  26 
793  ,  84 
273  ,  42 

2264  ,  51 

(3) 

Ds.  cts, 
273  45 


846 
283 
846 
674 
273 


37 
75 
91 
75 
25 


Ds.  cts. 

437  ,  62i 
386  ,  814 
243  ,  18| 

427  ,  37* 

428  ,  12i 


One  halfis  two-fourths;  and  one  half  more  make 
four  fourths,  and  three  fourths  more  make  seven 
fourths,  and  one  fourth  more  make  eight  fourths,  and 
one  half  (or  two  fourths)  more  make  ten  fourtiis. 
Four  fourths  make  one  cent,  then  ten  fourtiis  make 
two  cents,  and  leave  two  fourths,  or  one  half  cent. 
Set  down  the  i  cent,  and  carry  the  two  cents  to  the 
next  column. 


1923  ,  12* 


Ds.  cts. 
274  ,  814 
362  ,  87* 
421  ,  184 
625  ,  314 
241  ,  561 


Ds.  cts. 

27  ,  68| 

36  ,  81* 

28  ,  62i 

37  ,  934 
24  ,  62i 


Ds.  cts. 
56  ,  064 
32  ,  12i 
36  ,  25 

42  ,  62i 
54  ,  814 


32  FEDERAL   MONEY. 

APPLICATION. 

1.  Add  48  dollars  20  cents;  14  dollars  58  cents;   100 
dollars  25  cents;  and  84  dollars  36  cents. 

Ans.  247  dollars  39  cents. 

2.  Add  $7,62*,   $34,314,   $72,064,   $41,314,   $25, 
68|,  and  $87,43|  together,  and  tell  the  amount. 

Ans.  $268,43|. 

3.  Bought  a  hat  for  $4,25  cents ;   a  pair  of  shoes  for 
$2,25;  a  pair  of  stockings  for  $1,25,  and  a  pair  of  gloves 
for  75  cents.     What  is  the  cost  of  the  whole  ? 

Ans.  $8  50  cents. 

4.  If  I  buy  coffee  for  $1,18|,  tea  for  $2,50,  cloves 
for  87 ft,  mace  for  93|,  cinnamon  for  $l,87i,  raisins  for 
$2,68|,  nutmegs  for  37i,  candles  for  87£,  and  wine  for 
$1,93|,  what  must  I  pay  for  them?          Ans.  $13,25. 

Questions. 

What  relation  do  mills,  cents,  dimes,  tec.,  bear  to  each 
other? 

How  are  the  addition,  subtraction,  multiplication,  and 
division  of  Federal  money  performed? 

How  do  you  place  the  numbers  to  be  added? 

How  do  you  proceed  when  halves, fourths,  &,c.,  occur? 

SUBTRACTION  OF  FEDERAL  MONEY. 

RULE. — Place  the  less  under  the  greater,  with  dollars 
under  dollars,  and  cents  under  cents;  then,  if  there  are 
no  fractions,  proceed  as  in  simple  subtraction. 

If  there  is  a  fraction  in  the  upper  sum  and  none  in 
the  lower,  set  it  down  as  a  part  of  the  remainder,  and 
proceed  as  before. 

If  there  is  a  fraction  in  each  sum,  and  the  lower  be 
less  than  the  upper,  subtract  the  lower  from  the  upper, 
and  set  down  the  difference. 

If  the  lower  fraction  be  greater  than  the  upper  one, 
borrow  one  cent,  and  call  it  four  fourths,  and  add  them 
to  the  upper  fraction,  and  subtract  the  lower  one  from 
the  sum. 

Proof. — As  in  simple  subtraction. 


FEDERAL    MONEY.  33 

EXAMPLES. 

Ds  cts.  Ds.  cte.  Ds.  cts. 
32  ,62  43  ,  68|  75  ,  68* 
21  ,31  21  ,  25  24  ,  12i 

$11  ,  31      $22  ,  43|       51  ,  564 

NOTE. — TJiree  fourths  cannot  be  taken  from  two 

Ds.       cts.  fourths :  then  borrow  one  cent  from  the  two  cents, 

271       62*  which  has  four  fourths  in  it:  add  the/our  fourths  to 

,  or>  '   QQ  3  tne  two  fourths,  this  makes  six  fourths ;    subtract 

lo/5  ,  Jof  three  fourths  from  six  fourths,  and  three  fourths  (|) 

remain.     Set  down  the  £  and  add  one  to  the  next 

138  ,  68|  3,  as  in  simple  subtraction. 

EXERCISES. 

Ds.  cts.  Ds.  cts.  Ds.  cts. 
65  ,  49  520  ,  314  436  ,  31* 
35  ,  12i  210  ,  12*  243  ,  18| 


Ds.  cts.  Ds.  cts.  Ds.  cts. 
273  ,  62*  237  ,  564  732  ,  314 
124  .  37*  142  ,  874  261  ,  68| 


APPLICATION. 

1.  Subtract  $432,68|  from  1000,93|. 

Ans.  $568,25. 

2.  Subtraction    shows   the   difference   between  two 
numbers;  what  is  the  difference  between  $37,62 *  and 
$93,87*.  Ans.  $56,25. 

3.  Bought  goods  to  the  amount  of  $545,95,  and  paid 
at  the  time  of  purchase  $350;     How  much  remains  un- 
paid? Ans.  $195,95. 

4.  A  merchant  bought  a  quantity  of  cotil?e,  for  which 
he  paid    $560.     He  afterwards  sold  it  foi    $610,87* 
How  much  did  he  gain  by  the  transaction  ? 

Ans.  $50,87*. 

n   •> 


34  FEDERAL   MONEY. 

Questions. 

How  do  you  place  the  numbers  in  subtraction  of 
Federal  Money  ? 

How  do  you  perform  the  operation? 

If  a  fraction  occur  in  the  upper  line  or  minuend,  what 
do  you  do  with  it? 

If  a  fraction  occur  in  each,  how  do  you  proceed  ? 

Suppose  the  lower  fraction  is  greater  than  the  upper 
one,  how  do  you  proceed  ? 

How  do  you  prove  subtraction  of  Federal  Money  ? 

MULTIPLICATION  OF  FEDERAL  MONEY. 

RULE. — Set  the  multiplier  under  the  multiplicand, 
and  if  there  be  no  fractions,  proceed  as  in  simple  multi- 
plication; observing  to  separate  the  cents  from  the  dollars 
in  the  product. 

If  there  is  a  fraction  in  the  sum,  multiply  it,  and  see 
how  many  cents  are  in  the  product;  set  down  the  frac- 
tion that  is  over,  and  proceed  as  before. 

Or  if  the  multiplier  exceeds  12,  multiply  the  sum, 
omitting  the  fractions;  then  multiply  the  fraction,  and 
add  the  number  of  cents  contained  in  the  product,  to  the 
product  of  the  rest  of  the  sum. 

EXAMPLES. 

Ds.    cts. 
10  ,  56* 
2 

$50  ,  00  $21  ,  12*  $118  ,  12* 

Ds.  cts.  Ds.  cts. 

10,87*       125     times  4,18*      24  times  *  are 

125   one  half  make  24    72  fourths:    four 

125  halves:   2     fourths    are   con- 

5435   into  125  go  62  1672tained  18  times  in 

2174     times,  leaving  836    72  fourths,  rnak- 

1087       one;     that    is,  18  ing  18  cents. 

62*  one  half,  mak-     

-ing  62*  cents.  $100,50 

$1359,37* 


FEDERAL    MONEY. 
EXERCISES. 


1  Multiply 

2 

3 

4 

5 

G 


$145,18| 

7,874 

28,684 

42,314 

137,62* 

79,004 


by 
by 
by 
by 
by 
by  207 


7 

47 
68 
58 
67 


35 


Ans.  $1016,314 

370,124 

1950,75 

2454,124 

9220,874 

16354,034 


APPLICATION. 


1.  What  will  8  pounds  of  cheese  come  to,  at  18  cents 
a  pound?  Ans.  $1  44  cts. 

2.  What  is  the  value  of  12  yards  of  linen,  at  35  cents 
;a  yard?  Ans.  $4  20  cts. 

3.  What  cost  29  yards  of  cloth  at  $2  25  cts.  a  yard? 

Ans.  $65  25  cts. 

4.  What  wMl  213  barrels  of  flour  cost,  at  $5  25  cents 
a  barrel?  Ans.  $1118  25  cts. 

5.  Bought  321  barrels  of  cider  at  $1  25  cts.  a  barrel. 
What  did  it  amount  to?  Ans.  $401  25  cts. 

6.  What  will  580  bushels  of  salt  cost  at  $1  124  cts.  a 
bushel..  Ans.  $652  50 cts. 

7.  What  is  the  value  of  2  pieces  of  cloth,  one  contain- 
ing 38  yards,  and  the  other  26  yards,  at  $3  874  cts.  a 
yard?    '  Ans.  $248. 

8.  What  will  be  the  cost  of  132  pieces  of  linen  at 
$17  374  cts.  each?  Ans.  $2293  50  cts. 

9.  What  will  8  cords  of  wood  amount  to,  at  4  dollars 
{  50  cents  a  cord?  Ans.  36  dollars. 

10.  Sold  213  barrels  of  flour  for  6  dollars  25  cents  per 
barrel.     What  is  the  amount?  Ans.  1331  dols.  25  cts. 

11.  Bought  £)08  pounds  of  coffee  at  21  cents  a  pound. 
What  is  the  amount?  Ans.  64  dols.  68  cts. 

12.  Bought  217  gallons  of  brandy  at  $1  18|  cts.  per 
gallon;  and  sold  it  for  $1  374  cts.  per  gallon.  What  was 
the  amount  paid  for  the  whole;  the  sum  it  sold  for;  and 
the  gain? 

Ans.  Prime  cost,  $257  68|:  sold  for  $298  374;  gain, 
$40,681 


36 


FEDERAL  MONEY 


DIVISION. 

RULE. — Divide  as  in  simple  division.  When  a  re- 
mainder occurs,  multiply  it  by  4;  and  add  the  number 
of  fourths  that  are  in  the  fraction  of  the  sum  (if  any)  to 
its  product:  divide  this  product  by  the  divisor,  and  its 
quotient  will  be  fourths,  which  annex  to  the  quotient. 

Proof. — As  in  simple  division. 


Ds.  cts. 

2)45,22 


22,61 
Ds.  cts. 

25)629,68|(25,1S| 
50 


EXAMPLES. 

Ds.  cts. 
3)63.181 

21,064 


Ds.  cts. 
2)25,374 

12,684 

32)78800(24,624 
64 


129  148 

125  128 

46  200 

25  192 

218  80 

200  64 

18            Here  18  cents  remain;  multiply  16 

4         18  cents    by  four,  brings  them    to  4 
fourths  of  a  cent;  add  the  |,  this 
makes  75  fourths:  divide  75  fourths 

by  25,  and  £  are  obtained,  whicii  3*2/4(2  or  4 

25)7 ''(<*    place  in  the  quotient.  64 


Divid» 


D.  cts. 

56,15 

96,00 
156,00 

58,14 
417,96 
494,45 
627,38 


by 
by 
by 
by 
by 
by 
by 


EXERCISES. 

10 
5 

4 

38 

129 

341 

508 


Quotient  §  5,61  4 

-  19,20 

-  39,00 

-  1,53 
--      3,24 


1,234 


FEDERAL  MONEY.  37 

APPLICATION. 

1.  If  7  pounds  of  butter  cost  $1,89  cts.,  what  is  the 
value  of  1  pound?  Ans.  27  cts. 

2.  If  8  Ibs.  of  coffee  cost  $2,04  cts.,  what  is  the  price 
of  one  pound?  Ans.  25i  cts. 

3.  Bought  29  yds.  of  fine  linen  for  $65,25  cts.,  what 
was  the  price  per  yard?  Ans.  $2,25. 

4.  Paid  $58,75  cts.  for  235    yds.  of  muslin,  what 
was  it  per  yard?  Ans.  25  cts. 

5.  A  piece  of  cloth  containing  72  yds.  cost  $450, 
what  was  it  per  yard?  Ans.  $6,25. 

Questions. 

How  do  you   perform  division  of  Federal  Money? 
How  do  you  proceed  when  a  remainder  occurs? 


PROMISCUOUS  EXERCISES  IN  THE  PRECEDING  RULES. 

1.  Bought  18  barrels  of  potatoes,    each  containing 
3  bushels,  at  25  cts.  a  bushel,  what  did  they  cost? 

Ans.  $13,50. 

2.  A  farmer   sold   30   bushels  of  rye   at  87  cts.  a 
bushel,  30  bushels  of  corn  at  53  cts.  a  bushel;  8  bushel 
of  beans  at  $1,25  cts.   a   bushel;  2  yoke  of  oxen  at 
$62  a  yoke;  10  calves  at  $4  a  piece;  15  barrels  of 
cider  at  $2,37i  a  barrel,  what  was  the  amount  of  the 
whole?  Ans.  $251,62*. 

3.  What  will  be  the  price  of  four  bales  of  goods,  each 
bale  containing  60  pieces,  and  each  piece  49  yards,  at 
374  cents  a  yard?  Ans.  $4410. 

4.  Add  $324,43*  cts.  $208,09*  cts.  and  $507,90*  cts. 
together,  and  divide  the  sum  by  2,  and  what  will  be 
the  result?  Ans.  $520,21|. 

5.  Divide  400  dollars,  equally,  among  20  persons. 
What  will  be  the  portion  of  each  person?         Ans.  $20. 

6.  Divide  1728  dollars,  equally  among  12  persons. 
What  does  each  one  of  them  share?  Ans.  $144. 

7.  If  240  bushels  cost  420  dollars;  what  is  the  cost 
of  one  bushel  at  the  same  rate?  Ans.  $1.75. 


38  REDUCTION.        ' 

REDUCTION. 

REDUCTION  is  the  changing  of  a  sum,  or  quantity, 
from  one  denomination  to  another,  without  altering  the 
value. 

CASE  1. 

To  reduce  a  sum,  or  quantity,  to  a  lower  denomination 
than  its  own. 

RTTLE. — Multiply  the  sum,  or  quantity,  by  that  num- 
ber of  the  lower  denomination  which  makes  one  of  its 
own. 

If  there  are  one  or  more  denominations  between  the 
denomination  of  the  given  sum,  and  that  to  which  it  is 
to  be  changed,  first  change  it  to  the  next  lower  than  its 
own;  then  to  the  next  lower,  and  so  on  to  the  deno- 
mination required. 


DRY  MEASURE. 

TABLE. 

2  pints  (pts.)  make  1  quart,     qt. 

8»quarts  -  1  peck,      pc. 

4  pecks  -  1  bushel,  bu. 

NOTE. — This  measure  is  used  for  measuring  grain, 
salt,  fruit,  &,c. 

EXAMPLES. 

NOTE. — 1.  To  reduce  bushels  to  pecks,  multiply  by 
4,  because  each  bushel  has  4  pecks  in  it. 

1.  Reduce  23  bushels  to  pecks. 

bit. 

23 

4 

Amt.  92  pecks. 

2.  Reduce  35  bushels  to  pecks.        Amt.  140  peeks. 
NOTE. — 2.  To  reduce  pecks  to  quarts,  multiply  by  8, 

because  each  peck  has  8  quarts  in  it. 


REDUCTION.  39 

3.  Reduce  27  pecks  to  quarts. 


8 

Amt.  216  quarts. 

4.  Reduce  43  pecks  to  quarts.      Amt.  344  quarts. 
NOTE.  —  3.  To  reduce  quarts  to  pints  multiply  by  2, 

because  each  quart  has  2  pints  in  it. 

5.  Reduce  43  quarts  to  pints. 

qt. 

43 

2 

Amt.  86  pints. 

6.  Reduce  32  quarts  to  pints.  Amt.  64  pints. 
Reduce  34  bushels  to  pints.    . 

fctt. 
34 
4         Multiply  the  bushels  by  4  to  bring 

them  to  pecks. 
136 

8        Multiply  the   pecks  by  8   to  bring 

-  them  to  quarts. 
1088 

2        And  multiply  the  quarts  by  2  to  bring 

-  them  to  pints. 
Amt.  2176  pints. 


7.  Reduce  56  pecks  to  pints.  Amt.  896  pints- 

8.  Reduce  47  bushels  to  quarts.  Amt.  1504  qt 

9.  Reduce  85  bushels  to  pints.  Amt.  5440  pt 

10.  Reduce  63  pecks  to  quarts.  Amt.  504  qt. 

11.  Reduce  132  bushels  to  quarts.         Amt.  4224  qt. 

12.  Reduce  234  bushels  to  pints.         Amt.  14976  pt. 
NOTE. — 4.  When  several  denominations  occur,  reduce 

the  highest  denomination  to  the  next  lower  one,  and  this 
again  to  the  next  lower,  and  so  on ;  observing  to  add 
the  amount  of  each  denomination,  the  number  there  is 
of  that  denomination  in  the  given  sum. 


40  REDUCTION. 

EXAMPLES. 

1.  Reduce  23  bushels,  3  pecks,  5  quarts,  1  pint,  to 
pints. 

bu.   pe.   qt.    pt. 
23-3-5-1 

4  Multiply  the  bushels  by  4 

—  to  bring  them  to  pecks,  and 

92  add  the  3  pecks  to  the  amount, 

3  which  makes  95  pecks. 


Multiply  the  pecks  by  8  to 
bring  them  to  quarts,  and  add 
the  5  quarts,  which  makes 
765  quarts. 


Multiply  the  quarts  by  2 
to  bring  them  to  pints,  and 
add  the  1  pint  which  makes 
1531  pints. 


1531  amt. 


Or  thus: 
bu.  pe.  qt. 
23  3  -  5  - 

4  Multiply  by  4  as  above; 

—  add  the  3,  and  set  down  the 

95  amount,  &c. 

8 

765 
2 

1531  Amt.  as  before. 

EXERCISES. 

1.  Reduce  13  bushels,  2  pecks,  7  quarts,  1  pint  to 
pints.      •  Amt.  879  pints. 


REDUCTION.  41 

2.  Reduce  24  bushels,  3  pecks,  1  quart  to  quarts. 

Amt.  793  qt. 

3.  Reduce  7  bushels,  3  pecks  to  quarts.  Amt.  248  qt. 

4.  Reduce  3  pecks,  2  quarts  to  pints.      Amt.  52  pt. 

5.  Reduce  7  quarts,  1  pint,  to  pints.         Amt.  15  pt. 

6.  Reduce  32  bushels,  0  pecks,  1  quart  to  pints. 

Amt.  2050. 
7*  Reduce  5  bushels,  1  peck,  0  quarts,  1  pint  to  pints. 

Amt.  337  pt. 
8.  Reduce  43  bushels,  1  peck  to  pints. 

Amt.  2768  pt. 

Question*. 

What  is  reduction? 
For  what  is  case  first  used  ? 

How  do  you  reduce  a  sum  to  a  lower  denomination 
than  its  own? 

How  do  you  reduce  bushels  to  pecks? 

Why  do  you  multiply  by  4  ? 

How  do  you  reduce  pecks  to  quarts? 

Why  do  you  multiply  by  8? 

How  do  you  reduce  quarts  to  pints? 

How  do  you  reduce  bushels  to  pints? 


AVOIRDUPOIS  WEIGHT. 

TABLE. 

16  drams  (dr.)  make  1  ounce,  oz. 

16  ounces  1  pound,  Ib. 

28  pounds  1  quarter  of  a  cwt.  qr. 

4  quarters,  (or  112  lb.)*  1  hundred  weight,  cwt 

20  hundred  weight  1  ton,  T. 

NOTE. — By  this  weight  are  weighed,  tea,  sugar,  cof- 
fee, flour  and  other  things  subject  to  waste,  and  all  the 
metals,  except  silver  and  gold. 


*  The  gross  hundred  weight  of  112  pounds  is  nearly  out  cf  use: 
the  decimal  hundred  weight  of  100  pounds  is  taking  its  place. 


42 


REDUCTION. 


EXAMPLES. 

1.  Reduce  23  tons  to  hundred  weight. 

tons. 
23 
20 

Amt.  460  cwt. 

2.  Reduce  34  hundred  weight  to  quarters. 

cwt. 

34 

4 

Amt.  136  quarters. 

3.  Reduce  42  quarters  to  pounds,      qrs. 

42 

28 


336 

84 


Amt.  1176  pounds. 

4.  Reduce  73  pounds  to  ounces. 

Ibs. 

73 

16 

438 
73 

Amt.  1168  ounces. 

5.  Reduce  54  ounces  to  drams. 

oz. 
54 

16  t 

324 
54 

Amt.  864  drams. 


REDUCTION.  43 

6.  Reduce  35  tons  to  drams. 

tons. 
35 
20 

700  cwt. 
4 


2800  qr. 

28 

22400 
5600 

78400  Ib. 
16 

470400 
78400 

1254400  oz. 


7526400 
1254400 

Amount.  20070400  drams. 


EXERCISES. 

7  Reduce  24  pounds  to  drams.    Amt.  6144  dr. 

8  Reduce  36  hundred  weight  to  pounds. 

Amt.  4032  Ib. 

9.  Reduce  73  quarters"  to  ounces.     Amt.  32704  oz. 
10.  Reduce  2  tons  to  pounds.  Amt.  4480  Ib. 

11    Reduce  4  tons  to  drams.  Amt.  2293760  dr. 


44  REDUCTION. 

12.  Reduce  3  tons,  13  cwt.,  2  qu.,  14  Ibs.,  to  pounds. 

T.  cwt.  qr.  Ib. 
3-13-2-14 
20 

60  Or  thus: 

13  T.  cwt.     qr.    Ib. 

—  3-13-2-14 

73  20 

4  — 

73 

292  4 

2  

294 

294  28 

28  

2366 

2352  588 

588  

8246  pounds 

8232 

14 

8246  pounds. 

13.  Reduce  2  tons.  15  cwt.  2  qr.  to  quarters. 

Amt.  222  qr. 

14.  Reduce  3  tons.  25  Ib.  to  pounds,      ^nt.  6745  Ib. 

15.  Reduce  5  cwt.  3  qr.  14  Ib.  to  ounces. 

Amt.  10528  oz. 

16.  Reduce  2  cwt.  2  qr  14  ounces  to  drams. 

Amt.  71,904  dr. 

TROY  WEIGHT. 

TABLE. 

24  grains  (gr.)  make  1  pennyweight,  dwt. 

20  pennyweights    -  1  ounce,  oz. 

12  ounces  -  1  pound,  Ib. 

NOTE. — By  this   weight,  jewels,  gold,  silver,  and 
I  liquors,  are  weighed. 


REDUCTION.  45 

EXASEPLES. 

1.  Reduce  32  pounds  to  ounces.  Ib. 

'     32 
12 

Amt.  384  ounces. 

2.  Reduce  23  ounces  to  pennyweights.         oz. 

23 
20 

Amt.  460  dwt. 

3.  Reduce  43  pennyweights  to  grains,     dwt. 

43 
24 


172 

86 


Amt  1032  grains. 

4.  Reduce  53  pounds  to  grams.  Ibs. 

53 
12 


50880 
25440 


Amt.  305280  grains. 

EXERCISES. 

1.  Reduce  24  ounces  to  grains.  Amt.  11520  gr. 

2.  Reduce  32  pound*  to  pennyweights.  Amt.  7680  dwt. 

3.  Reduce  132  pounds  to  ounces.          Amt.  1584  oz. 

4.  Reduce  234  ounce's  to  grains.       Amt.  112320  gr. 


46  REDUCTION. 

5.  Reduce  463  pounds  to  grains.     Amt.  2666880  gr. 

6.  Reduce  47  pounds,  10  ounces,  15  pennyweights  to 
pennyweights.  Amt.  11495  dv/t. 

7.  Reduce  5  pounds,  6  ounces,  4  pennyweights,  20 
grains  to  grains.  Amt.  31796  gr. 


APOTHECARIES  WEIGHT. 

TABLE. 

20  grains  (gr.)  make  1  scruple,  sc.  9 

3  scruples  -  1  dram-,  dr.  3 

8  drams  -  1  ounce,  oz.  3 

12  ounces  -  1  pound,  Ib. 

NOTE. — By  this  weight  apothecaries  mix  their  medi- 
cines, but  they  buy  and  sell  by  Avoirdupois  Weight. 

EXERCISES. 

1.  Reduce  32  pounds  to  ounces.  Amt.  384  oz. 

2.  Reduce  43  ounces  to  drams.  Amt.  344  dr. 

3.  Reduce  27  drams  to  scruples.  Amt.  81  sc. 

4.  Reduce  37  scruples  to  grains.         Amt.  740  gr. 

5.  Reduce  28  pounds  to  drams.          Amt.  2688  dr. 

6.  Reduce  36  ounces  to  scruples.         Amt.  864  sc. 

7.  Reduce  27  drams  to  grains.  Amt.  1620  gr. 

8.  Reduce  23  pounds  to  grains.     Arrt.  132480  gr. 

9.  Reduce  3  pounds,  5  ounces,  2  scruples  to  scru- 
ples. Amt.  986.  sc. 

10.  Reduce  7  ounces,  5  drams,  14  grains  to  grains. 

Amt.  3674  gr. 

11.  Reduce  27  pounds,  7  ounces,  2  drams,  1  scruple, 
2  grains,  to  grains.  Amt.  159022  gr. 

CLOTH  MEASURE. 

TABLE. 
4  nails  (na.)  make  1  quarter  of  a  yard,    qr. 

4  quarters         -  1  yard,  yd, 

3  quarters        -  1  Ell  Flemish,  E.  Fl 

5  quarters         -  1  Ell  English,  E.  E. 

6  quarters         -  1  Ell  French,  E.  Fr. 
NOTE. — By  this  measure  cloth,  tapes,  linen,  muslin, 

&c.,  are  measured. 


REDUCTION.  47 

EXERCISES. 

1.  Reduce  24  yards  to  quarters.  Amt.  96  qr. 

2.  Reduce  32  quarters  to  nails.  Amt.  128  na. 

3.  Reduce  27  yards  to  nails.  Amt.  432  na. 

4.  Reduce  46  Flemish  ells  to  quarters. 

Amt.  138  qr. 

5.  Reduce  27  English  ells  to  quarters. 

Amt.  135  qr. 

6.  Reduce  34  French  ells  to  quarters. 

Amt.  204  qr. 

7.  Reduce  45  Flemish  ells  to  nails.         Amt.  540  na. 

8.  Reduce  36  English  ells  to  nails.         Amt.  720  na. 

9.  Reduce  54  French  ells  to  nails.        Amt.  1296  na. 

10.  Reduce  13  yards,  3  quarters  to  quarters. 

Amt.  55  qr. 

11.  Reduce  3  quarters,  2  nails,  to  nails.     Amt.  14  na. 

12.  Reduce  24  yards,  2  nails  to  nails.       Amt.  386  na. 

13.  Reduce  13  E.  ells,  2  qrs.,  3  nails  to  nails. 

Amt.  271  na. 


LONG  MEASURE. 

TABLE. 

12  inches  (in.)  make  1  foot,  ft. 

3  feet  1  yard,  yd. 

5*  yards  -  1  Rod,  Pole,  or  Perch,    p. 

40  poles  1  Furlong. 

8  Furlong  1  Mile. 

3  Miles  1  League. 

60  Geographic,  orJ  ,   Hpjrrpp 
694  Statute  Miles     j 

NOTE. — This  measure  is  used  for  length  and  di»» 
tances. 

A  Hand  is  a  measure  of  four  inches,  and  is  used  in 
measuring  the  height  of  horses. 

A  Fathom  is  6  feet,  and  is  chiefly  used  in  measuring 
the  depth  of  water. 


REDUCTION. 


EXERCISES. 


1.  Reduce  23  leagues  to  miles.  Amt.  69  m. 

2.  Reduce  43  miles  to  furlongs.  Amt.  344  f. 

3.  Reduce  27  furlongs  to  poles.  Amt.  1080  p. 

4.  Reduce  56  poles  to  yards.  Amt.  308  yd. 

5.  Reduce  132  yards  to  feet.  Amt.  396  ft. 

6.  Reduce  76  feet  to  inches.  Amt.  912  in 

7.  Reduce  24  miles  to  poles.  Amt.  7680  p. 

8.  Reduce  32  furlongs  to  yards.  Amt.  7040  yd. 

9.  Reduce  86  poles  to  inches.  Amt.  1 7028  in. 

10.  Reduce  26  leagues  to  yards.        Amt.  137280  yd. 

11.  Reduce  52  miles  to  feet.  Amt.  274560*  ft. 

12.  Reduce  5  leagues  to  inches.         Amt.  950400  in. 

13.  Reduce  24  degrees  to  statute  miles.  Amt.  1668  m. 

14.  Reduce  12  miles,  3  furlongs,  25  poles  to  poles. 

Amt.  3985  po. 

15.  Reduce  14  leagues,  2  furlongs  to  poles. 

Amt.  13520  po. 

16.  Reduce  3  leagues,  2  miles,  6  furlongs,  18  poles  to 
yards. ,  Amt.  20779  yds. 


LAND,  OR  SQUARE  MEASURE. 


TABLE. 


144  square  inches  make 

9  square  feet 
304  square  yards 
40  square  perches  - 

4  roods 


square  foot, 
square  yard, 
square  perch, 
rood, 
acre, 


ft. 
yd. 

P- 
r. 


NOTE. — This  measure  is  used  to  ascertain  the  quan- 
tity of  lands,  and  of  other  things  having  length  and 
breadth  to  be  estimated. 

EXERCISES. 

1.  Reduce  27  acres  to  roods.  .        Amt.  108  r.  || 

2.  Reduce  53  roods  to  perches.    '  Amt.  2120  p.  |] 


REDUCTION.  49 

3.  Reduce  28  perches  to  square  yards. 

Amt.  847  sq.  yds. 

4.  Reduce  36  square  yards  to  square  feet.       324  ft. 
5    Reduce  27  square  feet  to  square  inches. 

Amt.  3888  in. 

6.  Reduce  34  acres  to  perches.  Amt.  5440  p. 

7.  Reduce  42  roods  to  square  yards. 

Amt.  50820  sq.  yds. 

8.  Reduce  24  square  perches  to  square  feet. 

6534  ft. 

9.  Reduce  32  roods  to  square  feet.     Amt.  348480  ft. 

10.  Reduce  23  acres  to  square  inches. 

Amt.  144270720  sq.  in. 

11.  Reduce  11  acres,  2  roods,  19  perches  to  perches. 

Amt.  1859  p. 

12.  Reduce  17  acres,  3  roods  to  perches. 

Amt.  2840  p. 

13.  Reduce  12  acres,  12  roods,   12  perches  to  square 
yards.  Amt.  60863  sq.  yd. 

CUBIC,      OR  SOLID  MEASURE. 

TABLE. 

1728  cubick  inches  make  1  cubic    foot 

27  feet  1  cubic    yard 

40  feet  of  round  timber,  or )     ,   m 
50  feet  of  hewn  timber,     |     l  Ton  or  load 
128  solid  feet  1  Cord  of  wood 

NOTE. — This  measure  is  employed  in  measuring 
solids,  having  length,  breadth,  and  thickness  to  ba  esti- 
mated. 

EXERCISES. 

1  Re  Ace  29  cords  of  wood  to  cubick  feet. 

Amt.  3712  c.  i 

2  Reduce  32  cubic    yds.  to  feet.  Amt.  864  c.  f. 

3  Reduce  23  cubic    feet  to  inches.  Amt.  39744  c.  in. 

4  Reduce  32  cubic   yds.  to  inches.  Amt.  1492992  c.  in. 

5  Reduce  2  cords  of  wood  to  inches.  Amt.  442368  c.  in 

6  Reduce  3  cords,  10  feet  to  feet.  Amt.  394  f> 

7  .Reduce  1  cord.  3  feet,  136  inches  fo  inches. 

Amt.  226504  ta- 


50  REDUCTION. 

LIQUID  MEASURE. 

TABLE. 

4  gills  make  1  pint  pt. 

2  pints  (pts)  1  quart  qt. 

4  quarts  1  gallon  gal. 

42  gallons  1  tierce  te. 

63  gallons  1  hogshead  hhd. 

2  hogsheads  1  pipe  or  butt  pi. 

2  pipes  1  tun.  T — 

NOTE. — This    measure  is  employed  in   measuring 
cider,  oil,  beer,  &c. 

EXERCISES. 

1  Reduce  23  tuns  to  pipes.  Amt.  46  pi. 

2  Reduce  43  pipes  to  hogsheads.  Amt.    86  hhd. 

3  Reduce  34  hogsheads  to  gallons.  Amt.  2142  gal. 

4  Reduce  27  tierces  to  gallons.  Amt.  1134  gal. 

5  Reduce  53  gallons  to  quarts.  Amt.  212  qt. 

6  Reduce  724  quarts  to  pints.  Amt.  1448  pt. 

7  Reduce  37  pints  to  gills.  Amt.  148  g. 

8  Reduce  12  pipes  to  gallons.  Amt.  1512  gal. 

9  Reduce  4  hogsheads  to  quarts  Amt.  1008  qt. 

10  Reduce  32  gallons  to  gills.  Amt.  1024  g. 

11  Reduce  2  tuns  to  gills.  Amt.  16128  gills 

12  Reduce  32  gals  3  qts.  to  pints.  Amt.  262  pt. 

13  Reduce  2  hogsheads,  27  gals.  3  qts  to  quarts. 

Amt.  615  qt. 

14  Reduce  3  tons,  1  hogshead,  15  gals.  1  qt  to  pints. 

Amt.  6674  pt. 

MOTION,  OR  CIRCLE  MEASURE. 

TABLE.  * 

60  seconds  ("sec)  make  1  minuta              ^in. 

60  minutes  1  degree             °  deg. 

30  degrees  1  sine                  «,  sin. 

12  sines  (or  360  degrees)  1  revolution 

NOTE. — This  measure  is  employed  by  astronomers, 
navigators,  &c. 


REDUCTION.  51 

EXERCISES. 

1  Reduce  5  sines  to  degrees.  Ami.  150° 

2  Reduce  8  degrees  to  minutes.  Amt.  4801 

3  Reduce  6  minutes  to  seconds.  Amt.  360  sec. 

4  Reduce  12  sines  to  seconds.  Amt.  1296000  sec. 

5  Reduce  3  sines  15  degrees  to  minutes. 

Amt.  6300  min. 

TIME. 

TABLE. 

60  seconds  (sec)  make  1  minute         min. 
60  minutes  1  hour  H. 

24  hours  1  day 

7  days  1  week 

12  months  (or  365  days)  1  year. 

NOTE. — The  true  year,  according  to  the  latest  and 
most  accurate  observations,  consists  of  365  d.  5  h.  48  m. 
and  58  sec :  this  amounts  to  nearly  365$  days.  The  com- 
mon year  is  reckoned  305  days,  and  every  fourth  or 
leap  year  one  day  more  on  account  of  the  fraction  omit- 
ted each  year,  which  being  put  together,  every  fourth 
year  is  added  to  it,  making  leap  year  366  days. 
The  year  is  divided  into  12  months  as  follows. 

The  fourth,  eleventh,  ninth  and  sixth, 
Have  thirty  days  to  each  affixed, 
And  every  other  thirty-one, 
Except  the  second  month  alone, 
Which  has  but  twenty-eight  in  fine, 
Till  leap  year  gives  it  twenty-nine. 

OR  THUS: 

Thirty  days  hath  September, 
April,  June,  and  November, 
February  hath  twenty-eight  alone, 
And  each  of  the  rest  has  thirty  one. 
When  the  year  can  be  divided  by  four,  without  a  re- 
mainder, it  is  bissextile,  or  leap  year. 


52  REDUCTION. 

EXERCISES. 

1  Reduce  42  years  to  months.  Amt.  504  m. 

2  Reduce  23  days  to  hours.  Amt.  552  h. 

3  Reduce  36  hours  to  minutes.  Amt.  2160  min. 

4  Reduce  25  minutes  to  seconds.  Amt.  1500  sec. 

5  Reduce  14  days  to  minutes.  Amt.  20160  min 

6  Reduce  52  hours  to  seconds.  Amt.  187200  sec. 

7  Reduce  13  weeks  to  hours.  Amt.  2184  h. 

8  Reduce  12  weeks  to  minutes.  Amt.  120960  min. 

9  Reduce  3  years  to  minutes,  allowing  365  days  to 
each  year.    Amt.  1576800  min. 

10  Reduce  15  years  and  6  months  to  months. 

Amt.  186  m. 

11  Reduce  4  weeks,  3  days,  22  hours,  to  hours. 

Amt.  766  h. 

12  Reduce  7  years,  24  days,  43  minutes,  to  seconds. 

Amt.  222828180  sec. 

STERLING  MONEY. 

TABLE. 

4  farthings  (qr)  make    1  penny         d. 
12  pence  1  shilling       s. 

20  shillings  1  pound 

Farthings  are  usually  written  as  fractions  of  a  penny, 
thus-  i  one  farthing 

4  two  farthings  or  a  half  penny. 
I  three  farthings. 

EXERCISES. 

1  Reduce  14  pounds  to  shillings.  Amt.  280  s* 

2  Reduce  23  shillings  to  pence.  Amt.  276  d. 

3  Reduce  34  pence  to  farthings.  Amt.  136  qr. 

4  Reduce  4  pounds  to  pence.  Amt.  960  d. 

5  Reduce  13  shillings  to  farthings.  Amt.  624  qr. 

6  Reduce  16  pounds  to  farthings.  Amt.  15360  qr. 

7  Reduce  13  pounds  14  shillings,  to  pence. 

Amt.  3288  d. 

8  Reduce  3  pounds  15  shillings  6  pence  to  farthings. 

Amt.  3624  qr. 


REDUCTION.  53 

FEDERAL  MONEY. 

TABLE. 

10  mills  make  1  cent 
10  cents  1  dime 

10  dimes  1  dollar 

10  dollars          1  eagle 

EXERCISES. 

1  Reduce  5  eagles  to  cents.  Amt.  5000  ct. 

2  Reduce  3  dollars  to  mills.  Amt.  3000  m. 

3  Reduce  15  dimes  to  cents.  Amt.  150  ct. 

4  Reduce  3  eagles,  5  dollars  to  cts.  Amt.3500  ct. 

5  Reduce  7  dollars,  3  dimes,  6  cents,  to  mills. 

Amt.  7360  m. 

As  eagles,  dimes  and  mills  are  not  used  in  accounts, 
they  will  generally  be  omitted  in  the  subsequent  exer- 
cises of  this  work. 

4  fourths,  or  3  thirds,  or  2  halves,  make  1  cent. 
100  cents  -  1  dollar. 

6  Reduce  125  cents  to  halves  of  a  cent. 

Amt.  250  halves. 

7  Reduce  32  cents  to  fourths  of  a  cent. 

Amt.  128  fourths. 

8  Reduce  23  dollars  to  cents.    Amt.  2300  ct. 

9  Reduce  25  dollars  15  cents  to  cents.    Amt.  2515  ct. 

10  Reduce  15  dollars  374  cents  to  halves  of  a  cent. 

Amt.  3075  halves. 

11  Reduce  21  dollars  15  cents  to  thirds  of  a  cent. 

Amt.  6345  thirds. 

12  Reduce  5  dols.  37i  cents  to  fourths  of  a  cent. 

Amt.  2150  fourths. 

13  Reduce  15  dollars  33*  cts.  o  thirds  of- a  cent. 

Amt.  4600  thirds 

NOTE.  To  reduce  dollars  to  cenrs  annex  two  cyphers : 
thus  53  dollars  are  5300  cents. 

To  reduce  dollars  and  cents  to  cents,  place  them  to- 


54 lllIDUCTiOX. 

gather  without  any  separating  point,  and  the   amount 
will  be  cents.    Thus  35  dollars  24  cents  are  3524  cents. 

Questions. 

For  what  purpose  is  Dry  measure  used  ? 
For  what  is  Avoirdupois  weight  used? 

For  what  is  Troy  weight  employed? 

For  what  is  Apothecaries  weight  employed? 

For  what  is  Cloth  measure  employed? 

For  what  is  Long  measure  used? 

For  what  is  Land  or  Square  measure  used? 

For  what  is  Cubick -measure  employed? 

For  what  is  Liquid  measure  employed? 

For  wha.t  is  Sterling  currency  used? 

For  what  is  Federal  currency  used? 

CASE  2. 

To  reduce  a  sum  or  quantity  to  a  HIGHER  denomina- 
tion than  its  own 

RULE. — Divide  the  sum  or  quantity  by  that  number  of 
its  own  denomination  which  makes  one  of  the  denomina- 
tion to  which  it  is  to  be  changed. 

When  there  are  one  or  more  denominations  between  the 
denomination  of  the  given  sum  and  that  to  which  it  is 
to  be  changed;  first  change  it  to  the  next  higher  than  its 
own,  and  then  to4he  next  higher,  antl  so  on. 

Remainders  are  always  of  the  same  denominations  as 
the  sums  divided. 

DRY  MEASURE. 

EXAMPLES. 

1  Reduce  25  pints  to  quarts.  pts. 
•    NOTE.— Divide  by  2,  because  every  2  pints    2)25 
make  one  quart.     In  25  are  12  two's  and 

1  over,  that  is  12  quarts  and  1  pint.  ql.12 — Ipt 

2  Reduce  43  quarts  to  pecks.  qt. 
Divide  by  8,  because  every  8  qts.  make     8)43 

1  peck.     In  43  are  5  eights  and  3  over, 

that  is  5  pecks  and  3  quarts.  pe.  5 — 3qt 


^  REDUCTION.  55 

3  Reduce  2G  pecks  to  bushels.  bu. 
Divide  by  four  because  every  4  pecks     4)26 

make  1  bushel.    In  26  are  6  fours  and         

2  over;  that  is  6  bushels  and  2  pecks.      bu.  6 — 2  pecks 

4  Reduce  359  pints  to  bushels.  pi. 
Divide  pints  by  2,  brings  them     2)359 

to  quarts;  divide  quarts  by  8,  brings        

them  to  pecks,  and  divide  pecks  by     8)179 — 1  pt. 

4  brings  them  to  bushels.  

4)  22— 3qt 


5  b.  2  p.  3  qt.  1  pt. 

5  Reduce  81  quarts  to  bushels.   A.  2  bu.  2  pe.  1  qt. 

6  Reduce  134  pints  to  pecks.  8  pe.  3  qt. 

7  Reduce  194  pints  to  bushels,         3  bu,  0  pe.  1  qt. 

Questions, 

What  is  reduction? 

For  what  is  case  second  used  ? 

How  do  you  reduce  a  sum  to  a  higher  denomination 
than  its  own? 

When  there  are  one  or  more  denominations  between 
the  denomination  of  the  given  sum  and  the  one  to 
which  you  wish  to  reduce  it,  how  do  you  proceed  ? 

Of  what  denomination  is  the  remainder  always! 

How  do  you  bring  pints  to  quarts  ? 

How  do  you  bring  quarts  to  pecks? 

How  do  you  bring  pecks  to  bushels? 

How  do  you  bring  pints  to  bushels? 


AVOIRDUPOIS   WEIGHT. 

1  Reduce  65  cwt.  to  tons.  Result  3  tons  5  cwt. 

2  Reduce  27  quarters  to  cwt.  Res.  6  cwt.  3  qr. 

3  Reduce  109  pounds  to  qr.  Res.  3  qr.  25  Ib. 

4  Reduce  123  ounces  to  pounds.  Res.  7  Ib.  11  oz. 

5  Reduce  234  drams  to  ounces.         Res.  14  oz.  10  dr. 

6  Reduce  4274  drams  to  pounds.  Res.  16  Ib.  11  oz.  2  dr.  | 

7  Reduce  175  quarters  to  tons.     Res.  2  tons  3  cwt.  3  qr.  j 

8  Reduce  6745  pounds  to  tons.  Res.  3  tens  25  Ib.  i 


56  REDUCTION. 

TROY  WEIGHT. 

1  Reduce  378  ounces  to  pounds.   Result,  31  Ibs.  6  oz. 

2  Reduce  235  pennyweights  to  ounces. 

Res.    11  oz.  15  dwt. 

3  Reduce  748  grains  to  pennyweights. 

Res.  31  dwt.  4  grains. 

4  Reduce  678  pennyweights  to  pounds. 

Res.  2  Ibs.  9  oz  18  dwt. 

5  Reduce  732  grains  to  ounces.       -* 

Res.  1  oz.  10  dwt.  12  grains. 

6  Reduce  14752  grains  to  pounds. 

Res.  2  Ibs.  6  oz.  14  dwt.  16  gr. 

APOTHECARIES  WEIGHT. 

1  Reduce  432  ounces  to  pounds.  Res.  36  Ibs. 

2  Reduce  782  drams  to  ounces.     Res.  97  oz.  6  dr. 

3  Reduce  91  scruples  to  drams.  Res.  30  dr.  1  scr. 

4  Reduce  192  grains  to  scruples.  Res.  9  sc.  12  gr. 

5  Reduce  258  scruples  to  ounces. 

Res.  10  oz.  5  dr.  1  scr. 

6  Reduce  12660  grains  to  pounds. 

Res.. 2  lb.2  oz.  3  drs. 

CLOTH  MEASURE. 

1  Reduce  60  quarters  to  yards.  Res.  15  yds. 

2  Reduce  60  quarters  to  English  ell&. 

Res.  12  E.  ells. 

3  Reduce  60  quarters  to  French  ells. 

Res.  10  Fr.  ells. 

4  Reduce  60  quarters  to  Flemish  ells. 

Res.  20  Fl.  ells. 

5  Reduce  52  nails  to  quarters.  Res.  13  qr. 

6  Reduce  123  nails  to  yards.  Res.  7  yds.  2  qr.  3  na. 

7  Reduce  543  nails  to  English  ells. 

Res.  27  ells.  0  qr.  3  nails. 

LONG  MEASURE. 

1  Reduce  36  miles  to  leagues.  Res.  12  1. 

2  Reduce  75  furlongs  to  miles.  Res.  9  m.  3f. 


KEDUCTION.  57 

3  Reduce  295  poles  to  furlongs.  Res.  7f.  15  p. 

4  Reduce  286  yards  to  poles.  Res.  52  p. 

5  Reduce  365  feet  to  yards.  Res.  121  yds.  2  ft. 

6  Reduce  759  inches  to  feet.  Res.  63  ft.  3  in. 

7  Reduce  253  inciies  to  yards.  Res.     7  yds.  0  ft  1  inch. 

8  Reduce  2792  poles  to  leagues.  Res.    2  1. 2  m.  5  f.  32  p. 

SQUARE  MEASURE. 

1  Reduce  287  roods  to  acres.  Result  71  a.  3  r. 

2  Reduce  245  perches  to  roods.  Res.  6  r.  5  p. 

3  Reduce  756  square  feet  to  yards.  Res.  84  yds. 

4  Reduce  4731  square  yards  to  perches. 

Yds.  Res.  156  p.  12  yds. 


304 

4 


121 


18924 
121 

682 
605 

774 
726 


156  p. 


Bring  the  30^  yards  and  the  4731  yards 
both  to  fourths,  and  divide.  The  remainder 
48,  is  fourths  of  a  yard;  divided  by  four, 
brings  it  to  yards,  the  true  icmaiader. 


Rem. 


12  yards. 

5  Reduce  3575  square  inches  to  feet. 

Res.  24  feet  119  inches. 

6  Reduce  1728  square  perches  to  acres. 

Res.  10  a.  3  r.  8  p. 

CUBIC     MEASURE. 

1  Reduce  789  cubic    feet  to  cords.      Result,  0  c.  21  ft. 

2  Reduce  343  cubic    feet  to  yards.    Res.  12  yds.  19  ft. 

3  Reduce  9386  cubic    inches  to  feet.  Res.  5  "fi.  746  in. 

4  Reduce  70353i>  cubic    inches  to  cords. 

Res.  3  c.  23  ft.  243  in. 


c  2 


58  REDUCTION. 

LIQUID  MEASURE. 

1  Reduce  25  pipes  tq.tuns.  Res.  12  T.  1  P. 

2  Reduce  34  hogsheads  to  pipes.  Res.  17  P. 

3  Reduce  1575  gallons  to  hogsheads.          Res.  25  hhds. 

4  Reduce  163  quarts  to  gallons.  Res.  40  gal.  3  at. 

5  Reduce  6048  pints  to  tuns.  Res.  3  tuns. 

MOTION. 

1  Reduce  1440  seconds  to  minutes.         Result,  24  min« 

2  Reduce  720  minutes  to  degrees.  Res.  12  deg. 

3  Reduce  342  degrees  to  sines.      Res.  11  sines  12  deg. 

4  Reduce  443907  seconds  to  sines. 

Res.  4  sines  3  deg.  18  min.  27  sec. 

TIME. 

1  Reduce  1800  seconds  to  minutes.  Result,  30  m. 

2  Reduce  720  minutes  to  hours.  Res.  12h. 

3  Reduce  744  months  to  years.-  Res.  62  yrs. 

4  Reduce  4649  minutes  to  days.     Res.  3d.  5h.  29  min- 

5  Reduce  48888  minutes  to  weeks. 

Res.  4w.  5d.  22hrs.  48  min. 

STERLING  MONEY. 

1  Reduce  78  shillings  to  pounds  Res.3£.  18.9. 

2  Reduce  93  pence  to  shillings.  R^p.  7*.  9d. 

3  Reduce  39  farthings  to  pence.  Res.  JW.  3qr 

4  Reduce  656  pence  to  pounds.  Res.  2£  14s.  Sd. 

5  Reduce  781  farthings  to  shillings.     Res.  16s.  3d.  Iqr. 

6  Reduce  6529  farthings  to  pounds. 

Res.  6£.  16s.  OcZ.  1  qr. 

FEDERAL  MONEY. 

1  Reduce  250  halves  to  cents.  Res.  125  cts. 

2  Reduce  128  fourths  to  cents.  Res.  32  cts. 
Reduce  2343  cents  to  dollars.          Res.  23  dol.  43  cts. 

4  Reduce  15371  cents  to  dollars.     Res.  15  dol.  374  cts. 

5  Reduce  6150  half  cents  to  dollars.  Res.  30  dol.  75  cts. 
NOTE. — To  reduce  cents  to  dollars,  separate  two  figures 

01 1  the  right  hand  for  cents;  those  on  the  left  will  be  dollars. 


REDUCTION. 


59 


PROMISCUOUS    EXERCISES. 

1  How  many  bushels  in  738  quarts? 

Ans.  23  bushels  2  quarts. 

2  In  7  bushels,  how  many  pints?  Ans.  448  pints. 

3  How  many  cwt.  in  5356  ounces?  . 

Ans.  2  cwt.  3qr.  2Glb.  12  oz. 

4  How  many  drams  in  3  qr.  23  Ib.  14  oz.  ? 

Ans.  27610  drams. 

5  How  many  grains  in  9  oz.  14  dwt.  3  gr.? 

Ans.  4659  gr. 

6  How  many  pounds  in  7432  dwt.  ?Ans. 

301b.  lloz.  12  dwt. 

7  How  many  scruples  in  15  Ib.  1  oz.  6  drams? 

Ans.  4362  scr. 

8  How  many  ounces  in  218  scruples?  Ans.  9  oz. 

9  How  many  furlongs  in  2346  yards?  Ans. 

lOf.  26p.  3yd. 

10  How  many  poles  in  3  leagues?         Ans.  2880  poles. 

1 1  How  many  yards  in  84  nails?  Ans.  5  yd.  1  qr. 

12  How  many  perches  in  4719  square  yards 

Ans.  156  perches. 

13  How  many  square  yards  in  one  acre? 

Ans.  4840  sq.  yds. 

14  How  many  hogsheads  in  9728  gills? 

Ans.  4  hhds.  52  gal. 

15  How  many  pints  in  2  pipes?  Ans.  2010  pints. 

16  How  many  minutes  in  3  days  6  hours?   Ans.  4680m. 

17  How  many  hours  in  2  weeks  and  4  days? 

Ans.  432  hours. 

18  How  many  shillings  in  27  four  pences?          Ans.  9  s. 

19  How  many  cords  of  wood  in  9334  cubic  feet? 

Ans.  73  cords  20  feet.  ( 

.20  How  many  cubic  feet  in  9  cords?       Ans.  1152  feet.j 
21  How  many  inches  round  the  globe,  which  is  360  de- 
grees of  69i  miles  each?        Ans.  1,565,267,200  inches, 
Enumerate  the  answer. 


6t)  CO3IPOUXD    ADDITION. 


COMPOUND  ADDITION. 

COMPOUND  ADDITION  is  the  art  of  collecting  several 
numbers  of  different  denominations  into  one  sum. 


RULE. 
i 

Place  the  numbers  so  that  those  of  the  same  denom- 
ination may  stand  directly  under  each  other,  observing 
to  set  the  lowest  denomination  on  the  right,  the  next 
lowest  next,  &,c. 

Then  add  up  the  several  columns  beginning  with  the 
lowest  denomination :  divide  the  sum  by  as  many  of  the 
number  of  that  denomination  as  it  takes  to  make  one  of 
the  next;  and  so  on. 

Proof. — As  in  Simple  Addition. 


DRY  MEASURE. 

EXAMPLES. 

bu.       pe.       qt       pt.          The  first  column  on  the  right  makes 

3271       ^ve  PU)ts.     F 've   pints  make  two  quarts. 

and  leaves  one  pint.     Set  down   the  one 

pint  under  the  column  of  pints  and  carry 

the  two  quarts    to  the  column  of  quarts. 

6321      The  column  of  quarts  with  the  two  quarts 
2261      added  makes  twenty  four  quarts.     Twen- 
ty-four quarts  make  three  pecks  and  leave 

T~         ~ _      no  quarts.     Set  down  0  undei  the  column 
^*          <i  U  1      Of  quarts  and  carry  the  three  pecks  to  the 


column  of  pecks. 


The  column  of  pecks  with  the  three  pecks  added  makes  fourteen 
pecks.  Fourteen  pecks  make  three  bushels  and  leave  two  pecks. 
Set  down  the  two  pecks  under  the  column  of  pecks  and  carry  the 
three  to  the  column  of  bushels. 

The  column  of  bushels  with  the  three  bushels  added  makes 
twenty-four  bushels.  Here  set  down  the  whole  amount. 


COMPOUND    ADDITION.  61 

bu.  pe.  qt.  pt.  bu.  pe.  qt.      bu.  pe.  qt.  pt. 

23  371  437  371 

34  261  526  261 

42  351  615  050 

51  141  .704  141 

23  231  833  331 

14  1  2  1  412  021 

11  3  4  1  324  211 


202  3  2  1     40  3  7       3270 

APPLICATION. 

1  Add  2  bu.  3  pe.;7  bu.  3qt.;4  bu.  1  pe.  1  pt.;  6  bu. 
4  qt.  1  pt.;  and  3  pe.  1  qt.  together. 

Amount  21  bu.  0  pe.  1  qt.  0  pt. 

2  Add  3  bu.  2  pe.Sqt.  1  pt.;  7  bu.  7  qt.  1  pt.;  3  pe.  1  pt.; 
4  bu.  5  qt.;  4  bu.  3  pe.;  8  bu.  3  pe.  7  qt.  1  pt.  together. 

Amt.29  bu.  2  pe.  0  qt.  0  pt. 

3  Add  7  bu.  1  pt.;  3  pe.  7  qt.  1  pt.;  6  qt.  1  pt.;  9  bu.  3 
pe.  6  qt.  1  pt.;  3  bu.3  qt.;  4  bu.  1  pe.  Amt.  25  bu.  2  pe. 

4  In  a  wagon  load  of  grain  contained  in  seven  sacks, 
viz:  in  the  first  4  bu.  3  pe.  1  qt.;  in  the  second  5  bu.  7 
qt.  1  pt. — the  third  3  bu.  1  pe.  1  pt. — fourth,  3  bu.  2  pe. 
6  qt.— fifth,  5  bu. — sixth,  4  bu.  1  pe.  1  pt.: — and  in  the 
seventh  6  bu.  1  pe.  1  pt.     How  many  bushels? 

Questions. 

What  is  Compound  Addition? 

How  do  you  place  the  numbers  to  be  added? 

Do  you  place  the  greater  or  smaller  denominations 
in  the  right  hand  column? 

Where  do  you  begin  the  addition? 

When  the  first  column  is  added,  how  do  you  proceed 
with  the  sum? 

When  you  divide  the  sum  by  as  many  of  that  denom- 
ination as  make  one  of  the  next;  which  do  you  set  down, 
the  remainder  or  the  quotient? 

What  do  you  do  with  the  quotient? 


62  COMPOUND    ADDITION. 

In  what  particular  does  compound  addition  differ  from 
simple  addition? 

Do  you  carry  one  for  every  ten  in  compound  addition? 

Since  you  do  not  carry  one  for  every,  ten,  how  many 
do  you  always  carry?  A.  One  fo*  as  many  of  any  de- 
nomination as  make  one  of  the  next. 

Here  the  pupil  will  have  something  with  which  to  compare  simple 
addii-ion,  in  which  he  carries  one  for  every  ten.  This  comparison  will 
improve  and  correct  his  understanding  of  the  elementary  rules. 

AVOIRDUPOIS  WEIGHT. 

T.  cut.  qr.  Ib.  T.  cwt.  qr.  Ib.  oz.  dr. 

(1)  15  3  2  15  (2)  7  11  2  16  4  13 

4839  15  7  3  8  16  7 

82  19  1  10  13S  19  1  12  8  13 

163  8  3  17  42  8  3  19  12  4 

34  15  2  24  357  6  2  8  3  3 


300  16  1  19 

APOTHECARIES'  WEIGHT. 

£339  &  3  3  9    sr- 

(1)      6     3     L     2  (2)   84  7  t>  0   12 

19     9     5     1  132  5  0  2 

182     7     3    2  16  2  2  2    8 

57     6     1     0  1427  6  7  0  19 

40     5     0     0  14  0  6  1     9 


306     7     3     2 

TROY  WEIGHT. 

Ib.  oz.  dwt.  Ib.  oz.dwt.  gr. 

(1)    47  10  12  (2)     185  2  19  20 

38     8  6  56  9  15     6 

16  11  4  1472  11     2  17 

7    2  16  385  0    8     5 

13     9  11  10  8     7  12 


124    6     9 


C02IPOUND   ADDITION. 

63 

CLOTH  MEASURE. 

yds.  qr.  na.            E.E.  qr.  na. 

EFL  qr. 

na. 

(1)     75    3    2        (2)     72     3    2        (3)     19    2 

3 

163     13                536     2     1 

728     1 

2 

245     2     0                847     1     3 

142    0 

1 

738     3     1               1453     0     2 

,816     0 

0 

1785    23                 41     2     0 

32     1 

2 

3009     1     1 

LONG  MEASURE. 

L.  M.fur.  P.                                 yd.  ft.  iji. 
(1)     5    2    4     17                      (2)     3    2  11 

16     1     3     10                               1 

1     9 

72     0     5     24                               2 

0     8 

526     0     3     12                               3 

1   10 

834    2     6     34                               2 

0    4 

38     0     3     12                               6 

2     7 

1493     2     2     29 

SQUARE  MEASURE 

A.    R.    P.                        A. 

R.     P. 

(1)     39     2     37                    (2)    487 

2     17 

G2     1     17                               25 

3    28 

68     0    38                               67 

0    32 

129     3     12                               45 

1     16 

532     1     18                               26 

0    29 

832     2       2 

o< 

04 


COMPOUND    ADDITION. 


CUBIC     MEASURE. 


yd.  ft.  in. 

(I)     75  22  1412 

9  26  195 

3  19  1091 

28  15  1110 

49  24  218 

18  17  1225 


cords,  feet.     in. 

(2)     37  119  1015 

9  110       159 

48  127  1071 


8 

21 

9 


111       956 

9        27 
28     1091 


186     18 


67 


135       122       863 


3  In  four  piles  of  wood;  the  first  containing  32  feet  149 
inches;  the  second  121  feet  1436  inches;   the  third  97 
feet  498  inches;  the  fourth  115  feet  1356  inches;  how 
much  did  the  whole  amount  to? 

Ans.  2  cords,  110  ft.  1711  in. 

4  In  six  boat-loads  of  wood:  the  first  containing  22 
cords  114  feet,  987  in.;   the  second  18  cords,  121  feet, 
1436   in.;   the  third  21   cords,  109  feet,  1629  in.;   the 
fourth  15  cords,  82  £>et,  1321  in.;   the  fifth  16  cords,  98 
feet,  1111  in.;  the  sixth  24  cords,  89  feet,  987  in.     How 
much  did  they  contain? 

Ans.  120  c.  105  ft.  559  in. 


LIQUID  MEASURE. 

T.  hhd.  gal  lihd.  gal.  qt.  pt. 

(1)  18  2  54  (2)  385  42  3  1 

62  1  39  27  36  2  0 

327  0   4  132  17  0  0 

46  1  19  729  25  0  0 

285  3  28  173  47  2  1 


740  1  18 


COMPOUND   ADD/TION.  65 

MOTJOIV 

0  i       '  STNf  o  i  // 

(1)     17    55     48                     ^)  1  25  49  51 

1  37    51  2  4  21  36 
28     19    45  4  19  47  18 
19     19    37  1  25  25  39 


67     13       1  10     15    24    24 


3.  Add  5sm  10°  46' 38';  11°  37'  18";   Isin.   17°12' 
18";  Isin.  52";   Isin.  15°  12'  23";  and  11°  57' 29"  to- 
gether. Ans.  II  sin.  6°  46' 58". 

4.  Add  45';  Isin.  9°  18";  14°  21'  34";  2sin.  8°  13' 
54";  4sin.  7°  12'  19";  and  47'  32"  together. 

Ans.  Sain.  10°  20'  37". 

TIME, 

we.  d.  h.  H.  min.  sec. 

(2)  3     5  20  (3)  20     52    40 

2  3  17  k    122     12    35 

3  6  22  68       9     17 
0    4  16  135     37     12 
0    3  19  24     35    28 


231    7  11    3    22           371  7     12 

STERLING  MONEY. 

£    8.     d.  £    s.     d.  £    .?.     d. 

(1)     2    3    4  (2)     7     9    44         (3)     4    6    4 

712  13     7     6|  47  19    7 

973  452  159     53 

5    2    24  10  18  10$  78     6  11 


23  13  114 


4  Add  £703  Is.  4d.-,  £39  4*.  9<Z.;   £162  17s.  2d  ; 
£459  15*.;  £473  12*.  Sd  together. 

Ans.  £1898  16*.  lid. 


66  COMPOUND   SUBTRACTION. 

5  Add  the  following  sums :  viz.  £69  18s.  7d.;  £175 
2s.  Qd.;  £1582  19s  4cZ.;  £175  13s.  9d.;  £143  13s.  8d.-> 
and  £212  Os.  7d  Ans.  £2359  8s.  5d. 

COMPOUND  SUBTRACTION. 

COMPOUND  SUBTRACTION  is  'the  art  of  finding  the  dif- 
ference between  two  numbers  consisting  of  several  de- 
nominations. 

RULE. 

Place  the  numbers,  as  in  compound,  addition  with  the 
less  under  the  greater:  then  begin  at  the  right  hand 
denomination  and  subtract  the  lower  number  from  the 
upper,  and  set  down  the  remainder. 

If  the  upper  number  of  any  denomination  be  less  than 
the  lower  one,  add  to  the  upper  one  as  many  as  it  takes 
of  that  to  make  one  of  the  next;  subtract  the  lower  num- 
ber from  the  amount  and  set  down  the  remainder  as 
before. 

Proof. — As  in  simple  subtraction. 

EXAMPLES. 

bu.  pe.  qt.  pt  bu.  pe.  qt.  pt. 

7241  42     361* 

3120  31     231 


4121  11     1     3     0 


bush.  pe.       qt. 

925          We  cannot  take  7  quarts  from  5  quarts ;  then 
237      borsow  1  from  the  2  pecks.     One  peck  has  8 
__________^_      quarts  in  it:  8  quarts   added  to  the  5  quarts, 

f*  n  (*      make  13  quarts.     Take  7  qts.  from  13  qts.  and 

~  .6  qts.  remain.     Sot  down  the  6  qt. 

Because   I  borrowed  1  from  the  2  quarts,  I 

/nust  add  one  to  the  3  below  it,  which  makes  the  lower  figure  4.  Now 
[  pecks  from  2  pecks  we  cannot  take :  then  borrow  one  bushel  from  the 
I ;  that  bushel  has  4  pecks  in  it;  4  p.  and  2  p.  make  6  p.  Now  4  p. 
"rom  6  p.  and  2  pecks  remain,  which  set  down. 

Because  I  borrowed  1  from  the  9,  I  must  add  1  to  the  figure  below 
t.  1  to  2  make  3.  Take  3  from  9  and  6  remain.  Set  down  the  6, 
aid  the  work  is  done. 


COMPOUND  SUBTRACTION.  67 

bu.  pe.  qt.  pt.  bu.  pe.  qt.  pt. 
8271  8130 
4361  4251 


0 


bu.   pe.   qt.  bu.   pe.   qt.   pt. 

95    3    2  28    2    2    0 

22    0    1  14    3    5    1 


APPLICATION. 

1.  From  a  granary  containing  94  bushels,  2  pecks,  7 
quarts,  have  been  taken  43  bush.  3  pe.  5  qr.     How  much 
remains?  Ans.  50  bush.  3 pe.  2  qt. 

2.  From  a  wagon  load  of  corn  containing  63  bushels, 
3  pecks,  4  qts.,  have  been  sold  27  bush.  3  pe.  7  qt.  1  pt. 
How  much  remains  unsold?     Ans.  35  bu.  3  p.  4qt,  1  pt. 

Questions. 

What  is  compound  subtraction? 

In  what  particular  does  it  differ  from  simple  subtrac- 
tion? 

How  do  you  place  the  numbers  in  compound  subtrac  • 
tion? 

Where  do  you  begin  the  operation? 

When  the  upper  number  of  any  denomination  is  less 
than  the  lower  one,  how  do  you  proceed? 

Do  you  borrow  one  from  the  next? 

Do  you  call  the  number  you  borrow,  one  ten,  as  in 
simple  subtraction? 

What  do  you  call  it? 

Ans.  I  call  it  one  peck,  or  one  yard,  or  one  mile,  as 
the  case  may  be  ? 

What  do  you  do  with  it  then? 

Ans.  I  reduce  it  to  quarts,  or  to  feet,  or  to  furlongs, 
&c.  according  to  the  nature  of  the  case;  then  add  these 


68  COMPOUND   SUBTRACTION. 

to  the  upper  figure  on  the  right,  subtract  the  lower  figure 
from  the  sum,  and  set  down  the  remainder. 

When  you  borrow  one  from  the  upper  figure,  why  do 
you  add  one  to  the  figure  below  it? 

NOTE. — Upon  a  clear  conception  of  the  principles  involved  in  these 
questions,  depends  the  pupil's  correct  knowledge  of  the  science  of 
Arithmetic. 

AVOIRDUPOIS  WEIGHT. 
tons  cwt.  qr.  tons  civt.   qr.  Ib.     cwt.  qr.  Ib.  oz.  dr. 
From  45  11  3       52     12     3     15       17     0     0     0     0 
Take   15  10  2      24     10    0     26        6     3  21   15     9 

Rem.    30     1  1       28      2    2     17       10     0     6     0.   7 

1.  Subtract  76  tons,  18  hundred  weight,  3  quarters, 
from  195  tons,  2  hundred  weight,  2  quarters. 

Ans.  118  tons,  3  cwt.  3  qr. 

2.  Subtract   14  pounds,  6  ounces,  3  drams  from  20 
pounds,  2  ounces.  Ans.  5  Ibs.  11  oz.  13  dr. 

APOTHECARIES  WEIGHT, 

ft       g    3  ft     3     3    &  gr. 

1090     16  48     9     6     1  4 

106    2     7  1  10     0    2  8 


983  10     7 

3.  From  59ft  1£  23  take  53ffi  73  63.  Ans.  5fc  5g  5J. 

4.  Subtract  14fc  1>3  13  from  G9§.     Ans.  54ft  x>3  73. 

TROY   WEIGHT. 

Ib.  oz.  dwt.  gr.     Ib.  oz.  dwt.  gr.  Ib.  oz.  dwt.  gr. 

10     6  18     0        8     3     0    2  106  0    0     15 

4    0    2  20        2     1  18     6  10  6    2    20 


6    6  15  4 


4.  Subtract  I4lb.  Goz.  \\dwt.  from  92lb.  I2dwt.  6  gr. 

Ans.  lib.  Qoz.  Idwt.  63*r. 

5.  From  16Z&.  take  I2lb.  lloz.  iQdwt.  \\gr. 

Ans.   3Z&.  Ooz.  Qdwt.  I3gr. 


COMPOUND    SUBTRACTION.  60 

CLOTH  MEASURE. 

yds.  qrs.  na.        E.E.  qrs.  na.  E.  FL  qrs.  na 

From       71     3     1             42     0    2  51     2     2 

Take        14    2     3             19     2    3  42     2     1 


Rem.       57     0    2  22    2     3  901 

4.  Subtract  95  yards,  3  quarters,  2  nails,  from  156 
yards,  2  quarters,  3  nails.      Ans.  60  yds.  3  qr.  1  nail. 

5.  Subtract  14  English  ells,  1  quarter,  2  nails,  from 
52  English  ells,  3  quarters,  2  nails.    Ans.  38  yds.  2  qr. 

LONG  MEASURE. 

L.  M.fur.  L.  M.  fur.  P.  yds.  ft.  in. 

From  24     I     7  58     1     0     19  6     2     10 

Take    18     2    4  10     0     7    20  327 


Rem.     523  46    00    39  303 

4.  Subtract  45  miles,  5  furlongs,  20  poles,  from  320 
miles,  3  furlongs,  36  poles.      Ans.  274  M.  6  F.  16  P. 

5.  Subtract  15  yards,  2  feet,  6  inches,  from  36  yards, 
1  foot,  11  inches.  Ans.  20  yds.  2  ft.  5  inches. 

LAND,  OR  SQUARE  MEASURE. 
A.    R.   P.  A.   R.  P.          Yds.  ft.  in. 

From     96    3     36  195    22  25    2     72 

Take     25     2     39  36     3     1  14     7     10 


Rem.     Tl    0     37  158     3     1  10    4     62 

4.  Subtract  36  acres,  2  roods,  from  900  acres,  3 
roods,  16  perches.  864  A.  1  R.  16  P. 

5.  Subtract   72  acres,  from  360  acres,  2  roods,  29 
perches.  288  A.  2  R.  29  P. 

CUBICK  MEASURE. 

yds.       ft.          in.             cords,     ft.  in. 

79         11         917            349        97  1250 

17        25       1095             192       127  1349 


61         12       1550  156        97         1629 


70  COMPOUND   SUBTRACTION. 

1.  From  a  pile  of  wood  containing  432  cords,  27  feet, 
and  1432   inches,  have  been  hauled  156  cords,  92  feet, 
946  inches:  how  much  remains? 

Ans.  275  cords,  63  feet,  486  in. 

2.  From  a  bank  of  earth  containing  2984  yards,  18 
feet,  have  been  taken  143G  yards,  21  feet,   what  re- 
mains? Ans.  1547  yds.  24  feet. 

LIQUID  MEASURE. 

T.  hhd.gal.qt.pt.  T.  hhd.gal.qt.pt. 

2     3  50     1     0  100     1   19     2     1 

1     2  16     3     1  99     1  28     3     1 


1     1  33     1     1 

3.  If  I  purchase  2hhd.  of  wine,  and  to  oblige  a  friend 
send  him  29^aZ.,  what  quantity  have  I  left? 

Ans.   \hlid.  34gal. 

4.  Bought  1  pipe  of  wine,  4hhd.  of  brandy,  2  barrels 
of  beer;  I  have  since  sold  93  gallons  of  wine,  29  of 
brandy,  1  barrel  of  beer:   how  much  of  each  have  I 
remaining? 

Ans.   33gal.   of   wine,  223gal.   of   brandy,  and    1 
barrel  of  beer. 

MOTION. 

O  f  n  gin>      O  ,  ,/ 

79   21   31      6  10   12  48 
41   41  52      38   39  2 


37   39  39      3   1   33   19 

A  circle  being  12  sines,  how  far  has  the  hand  of  ? 
watch  to  pass,  after  having  gone  through  4  sines,  23' 
.5'  29"  ? 

Sin.    °        '      " 

12     0       0      0 

4  23    15    29 


COMPOUND   SUBTRACTION. 


71 


2  A  person  resioing  in  latitude  27C  32'  45"  north, 
wishes  to  visit  a  place  52°  24'  18'  north.  How  many 
degrees,  minutes,  and  seconds  northward  must  he 
travel?  Ans.  24°  51'  33'. 


TIME 

Y.  M.      w.  d.  ho.  min.  sec. 
69        3     1     3     40    20 
16        2    6    2     57     36 


H.min.sec.       Y.  M. 

16  29  33         18  11 

7  36  44          9  10 


53 


2     0    42    44 


4.  From  900  Y.  take  111F.  6m. 

Ans.  788  Y.  6m. 

5.  If  I  take  IF.  1M.  from  6Y.   what  space  of  time 
will  still  remain? 

Ans.  4F.  11  M. 

NOTE.  —  To  ascertain  the  amount  of  time  passed  be- 
tween two  events,  set  down  the  year,  month,  and  day  of 
the  latter  event,  and  place  those  of  the  former  below  it, 
and  subtract. 

6.  A  bond  was  given  24th  July,  (7th  month)  1809,  and 
paid  off  13th  August,  1821. 

yrs.  mo.  ds. 
1821  8  13 
1809  7  24 


12     0    20 

7.  The  declaration  of  independence  of  the  United 
States  passed  Congress,  4th  July,  (7th  month)  1776; 
and  the  declaration  of  the  late  war  with  Great  Britain, 
18th  June,  (6th  mo.)  1812.  How  many  years,  &,c.  be- 
tween them?  Ans.  35yr.  llmo.  14d. 

STERLING  MONEY. 

£.      ,.      d.          £     s.    d.          £      s.    d. 
146     19     lOi         47     6     71        419     7    6 


7     19      9| 
139       0       0| 


28 


lOi 


227     8     94 


72  COMPOUND   MULTIPLICATION. 

4.  Subtract  £200  9s.  from  £1000  Us. 

Ans.  £800  2s. 

5.  I  have  a  purse  of  money  containing   £1000  2s. 
4id.:  it  I  take  out  £60  7*.  8|d.  what  sum  will  be  left? 

Ans.  £939  14*.  7|d. 


COMPOUND   MULTIPLICATION. 

COMPOUND  MULTIPLICATION  is  the  art  of  multiplying 
numbers  composed  of  several  denominations. 

CASE  1. 
When  the  multiplier  does  not  exceed  12. 

RULE. 

Place  the  number  to  be  multiplied  as  directed  in 
compound  addition,-  and  set  the  multiplier  undes-the 
lowest  denomination. 

Multiply  as  in  simple  multiplication,  and  divide  the 
product  of  each  denomination  by  as  many  as  it  takes  of 
that  to  make  one  of  the  next  greater;  set  down  the  re- 
mainder (if  any)  and  carry  the  quotient  to  the  product 
of  the  next  denomination. 

Proof. — Double  the  multiplicand  and  multiply  by  half 
the  multiplier. 

EXAMPLES. 

Bu.        pe.        qt.     pt.          7  times  1  pint  make  7  pints ;  2  pt. 
7          2          5  1      make  1  qt. ;  then  7  pt.  make  3  qt.  and 

~  leave  1  pt.  Set  down  the  1  pt.  and 
carry  the  3  qt.  to  the  product  of  the 
next  figure. 


53  261  7  times  5  qt.  make  35  qt.  to  which 
add  the  3  qt.  which  make  38  qt. ;  8  qt.  make  one  peck ;  then  38  qt. 
make  4  pecks  and  leave  6  qts.  Set  down  the  six  quarts  and  carry  the 
4  pecks. 

7  times  2  pecks  make  14  pecks ;  add  the  4  pecks,  makes  18  pe. ; 
4  pecks  make  one  bushel ;  then  18  pe.  make  4  bushels  and  leave  2 
pecks.  Set  down  the  2  pecks  and  carry  the  4  bushels. 

7  times  7  bushels  make  49  bushels ;  add  the  4  bushels,  makes  53 
bushels,  which  set  down,  and  the  work  is  done. 


COMPOUND   MULTIPLICATION.  73 

Bu.      pe.      qt.       pt.  Bu.      pe.  qt.       pt. 

9361  23        2  5         1 

5  8 


49        3        0         1         189         1         4        0 

1.  In  one  vessel  are  contained  29  bushels  2  pecks  and 
5  quarts :  how  many  in  9  such  vessels  ? 

Ans.  266  bu.  3pe.5qt. 

2.  If  one  tub  will  contain  8  bu.  3  pe.  5  qt.  how  much 
will  11  such  tubs  contain?          Ans  97  bu.  3  pe.  7  qt. 

CASE  2. 

When  the  multiplier  exceeds  12,  and  is  the  exact  product 
of  two  factors  in  the  multiplication  table. 

RULE. 

Multiply  the  given  sum  by  one  of  the  factors,  and  the 
product  by  the  other  factor. 
Proof. — Change  the  factors. 

EXAMPLES. 

1.  Multiply  3  bushels,  2  pecks,  7  qt.  by  24.     Product. 
bu.       pe.       qt.  bu.       pe.      qt. 

327  327 

6  4 


22         1         2  14        3        4 

4  6 


89         1         0  Proof  89         1         0 

OR  THUS: 

bu.     pe.      qt.  bu.     pe.       qt. 

327  327 

3  8 


11         0         5  »o        3         0 

8  3 


89        1         0  89         1         0 

2.  Multiply  7  bushels,  3  pecks,  5  quarts,  by  36. 

Product,  284  bu.  2  pe.  4  qt. 

3.  Multiply  19  bushels,  2  pecks,  3  quarts,  bv  42 

D 


74  COMPOUND  MULTIPLICATION. 

CASE  3. 

When  the  multiplier  exceeds  12,  and  is  NOT  the  product 
of  any  two  factors  in  the  multiplication  table. 

RULE. 

Multiply  by  the  two  factors  whose  product  is  the  least 
short  of  the  given  multiplier;  then  multiply  the  given 
sum  by  the  number  which  supplies  the  deficiency;  and 
add  its  product  to  the  sum  produced  by  the  two  factors. 

EXAMPLES. 

1.  Multiply  21  bushels,  1  peck,  7  quarts,  by  23.  Prod. 

bu.   pe.  qt.  bu.    pe.  qt. 

21     1     7X3  21     1  '7x2 

5  3 


OR  THUS: 


107     1     3  64     1     5 

4  7 


429     1     4    product  of  20     450     3     3   product  of  21 
64     1     5    product  of  3        42    3    6    product  of  2 

493    3     1    product  of  23     493     3     1  product  of  23 

2.  Multiply  19  bushels,  3  pecks,  7  quarts,  by  34. 

Product,  678  bu.  3  pe.  6  qt. 

3.  Multiply  7  bushels,  3  pecks,  4  quarts,  by  59. 

4.  Multiply  9  bushels,  3  pecks,  2  quarts,  by  47. 

5.  Multiply  15  bushels,  1  peck,  7  quarts,  by  78. 

6.  Multiply  12  bushels,  2  pecks,  by  92. 

7.  Multiply  1?  bushels,  3  quarts,  by  98. 

8.  How  many  bushels  in  104  sacks,  each  containing 
7  bushels,  2  pecks,  3  quarts? 

9.  How   many   bushels  of  wheat  on  125  acres,  con- 
taining 21  bushels,  3  pecks  each? 


L. 


COMPOUND    MUI/TIPLICATIOX.  75 

CASE  4. 

Wlien  the  multiplier  is  greater  than  the  product  iff  any 
two  numbers  in  the  multiplication  table. 

RULE. 

Multiply  the  given  number  by  10,  as  many  times 
less  one  as  there  are  figures  in  the  multiplier. 

Multiply  that  product  by  the  left  hand  figure  of  the 
multiplier. 

Multiply  the  given  sum  by  the  units  figure  of  the 
multiplier;  the  product  of  the  first  10  by  the  tens  figure 
of  the  multiplier;  the  hundreds  product  by  the  hundreds 
figure  of  the  multiplier,  and  so  on,  till  you  have  multi- 
plied by  all  the  figures  of  the  multiplier  except  the  left 
hand  one. 

Add  all  the  products  together,  and  you  have  the  pro- 
duct required. 

EXAMPLES. 

1.  Multiply  3  bushels,  3  pecks,  1  quart,  by  45G. 
bu.    pc.     qt.  Product  1724  bu.  1  pe. 

3         3         1X6 

10  Because  there  are  3  figures,  multiply 

.  2  times  by  10.  Multiply  that  product  by 

ty-  o  v  E    the  kft  hand  figure  (4)  of  the  multiplier. 

^  X  °  Multiply  the  given  number  by  the  units 
figure  (6)  and  set  the  product  beneath. 
Multiply  the  10's  product  by  the  tens 


378  0  4  figure  (5)  of  the  multipliar. 

Add  the  several  products. 


1512  2  0 

22  2  6 

189  0  2 

1724  1  0  Product 


76 


COMPOUND   MULTIPLICATION. 


2.  Multiply  53  bushels,  2  pecks,  7  quarts,  by  2345. 
bu.      pe.         qt. 
53        2        7X5 
10 


537 


0        6+4 
10 


5371 


4+3 
10 


53718 


107437  2 

161 < 5  2 

2148  3 

268  2 


0  product  of  the  2000 

4  "            "      300 

0  "             «        40 

3  "            "5 


125970 


1 


7  Product  of  the    2345 


NOTE. — Let  the  pupil  try  experiments,  by  multiplying  simple  num- 
bers in  this  way. 

3.  Multiply  72  bushels,  1  peck,  2  quarts,  by  4723. 

Product,  341531  bu.  3  pe.  6  qt. 

4.  Multiply  13  bushels,  2  pecks,  4  quarts,  by  5124. 

Questions. 

What  is  compound  multiplication? 

In  what  does  it  differ  from  simple  multiplication? 

When  the  multiplier  does  not  exceed  12,  how  do  you 
proceed  ? 

How  many  do  you  always  carry  ? 

How  do  you  prove  compound  multiplication? 

How  do  you  proceed  when  the  multiplier  exceeds  12, 
and  is  the  exact  product  of  two  numbers  in  the  multi- 
plication table? 

When  the  multiplier  exceeds  12,  and  is  not  the  exact 
product  of  any  two  numbers  in  the  table,  how  do  you 
proceed? 


COMPOUND   MULTIPLICATION.  77 

How  do  you  proceed  when  the  multiplier  is  greater 
than  the  product  of  any  two  numbers  in  the  table  ? 

AVOIRDUPOIS  WEIGHT. 

tons.      cwt.      qrs.         cwt.      qr.       Ib.  oz.     dr. 

23         12        3  7        3         14  9        6 

4  6 


94         11         0  47         1         3        8 


tons     act.     qr.  cwt.      qr.       Ib.       oz.       dr. 

7         15        3  7        3        24       12         14 

8  9 


5.  Multiply  7  tons,  16  cwt.  3  qr.   by  24. 

Product,  188  T.  2  cwt. 

6.  Multiply  3  cwt.  2  qr.21  Ib.  14  oz.  by  30. 

Product,  110  cwt.  3  qr.  12  Ib.  4  oz. 

7.  Multiply  3  tons,  7  cwt.  2  qr.  by  34. 

Product,  114  tons  15  cwt. 

APOTHECARIES  WEIGHT. 

tt  g  3  9          fc   g   3  9  #••        fc  3  3  Bgr. 
4821          53   10  0  2  12         17  5  6  1  4 
5  9  12 


23  5  3  2 

TROY  WEIGHT. 

Ib.    oz.  dwt.        Ib.  oz.  dwt.  gr.        Ib.    oz.  diet.  gr. 
67     5     16          43     0    8     10         113     6     0    6 
246 


134  11     12 

4.  Multiply  41  Ib.  6  oz.  18  dwt.  2  gr.  by  7. 

Ans.  291  Ib.  0  oz.  6  dwt.  14  gr. 

5.  Multiply  91  Ib.  4  oz.  14  dwt.  16  gr.  Ly  8. 

Ans.  731  Ib.  1  oz.  17  dwt.  8  gi 

7» 


78  COMPOUND    MULTIPLICATION. 

CLOTH  MEASURE. 

yd.  qr.  na.     E.  E.  qr.  na.     E.Fl.  qr.  na.  E.Fr.  qr.  na. 

20    2    3       37     4     2          18    0    3  14     1     3 

6                    8                      12  9 


124     0    2 

5.  If  19  yd.  1  qr.  2  na.  be  multiplied  by  5,  what  num- 
ber of  yards  will  there  be  ?     Ans.  96  yds.  3  qr.  2  na. 

6.  Multiply  56  Ells  Eng.  3  qr.  by  9. 

Ans.  509  Ells  E.  2  qr. 

LONG  MEASURE. 

deg.m.fur.p.         1.   m.fur.p.          m. fur.  p.  yd.  ft.  in, 

8     1     3  36        4     2     2  29          18  3  20     1     2  10 

12  7  5 


96  17    6  32 

4.  Multiply  6  deg.  40  m.  7  fur.  by  10. 

Ans.  65  deg.  61  m.  2  fur. 

5.  Multiply  44  m.  6  fur.  20  p.   by  7. 

Ans.  313  m.  5  fur.  20  D 

LAND,  OR  SQUARE  MEASURE. 
a.     r.     p.  a.     r.    p.  a.     r.    p. 

49    2     17  19     3     20  10     0     33 

2  6  9 


99     0     34 

4.  How   many  acres  will  10  men  reap  in  one  day, 
allowing  them  1  acre  3  roods  11  perches  each? 

Ans.   18  A.  OR.  30  P. 

5.  Multiply  63  acres  3  roods  18  perches,  by  11. 

Ans.  702  A.  1  R.  38  P. 

6.  How  many  acres  in  15  lots,  containing  17  acres,  2 
roods,  and  20  perches  each?  Ans.  264 A.  1  R.  20P. 


COMPOUND   MULTIPLICATION.  79 


CUBIC  MEASURE. 

cords,  ft.      in.  yd.    fir     in. 

7    28     1327  19    23     1421 

6  8 


43    44     1050  159       1     1000 


cords,  ft.     in.  yd.    ft.     in. 

21     56     1432  27     13     1291 

7  9 


5  In  a  pile  of  wood  are  14  cords  92  feet;  how  much  in 
24  such  piles?  Ans.  353  c.  32  ft. 

6  In  a  cellar,  are  contained  42  yards  25  feet ;  what 
are  the  contents  of  23  such  cellars?     Ans.  987  yd.  8  ft. 

LIQUID  MEASURE. 

hhd.  gal.  qt.          T.hhd.gal.  qt.  pt.         pi.  hhd.gal.  qt.  pt. 

8    43     2          1     2  16     3     1          4     1  19    3     1 

4  '10  5 


34     48     0 

4  Multiply  3  T.  2  hhd.  50  gal.  2  qt.  by  8. 

Ans.  29  T.  2  hhd.  26  gal.  0  qt. 

5  Multiply  4  hhd.  41  gal.  1  pt.  by  10. 

Ans.  46  hhd.  33  gal.  1  qt.  0  pt. 

MOTION. 

sin.     °      '  sin.   °     '     " 

3    27    48  1  24  48  25 

7  9 


27     14    36  16  13  15  45 

3  If  a  planet  move  through  2sin.  15°  23'  of  its  orbit 
in  one  day ;  how  far  will  it  advance  in  8  days. 

Ans.  205m.  3°  4' 


80  COMPOUND    DIVISION 


TIME. 

weeks    rf.     h.  d.     h.     min.  sec. 

o    5    23  3     14    25    36 

8  9 


8  30    5     16  32       9    50    24 

4  If  a  man  can  perform  a  piece  of  work  in  2  yr.  3  mo., 
how  long  would  it  take  him  to  perform  5  such? 

Ans.  il  yr.  3  mo, 

5  If  a  laborer  dig  a  drain  in  2  weeks,  3  days,  how  long 
a  time  would  he  require  to  dig  9  such  drains? 

Ans.  21  weeks  6  days. 

STERLING  MONEY. 

£     s.    d.              £    s.     d.  £    s.     d. 

246  13    3|             14    6    04  111  11  10* 

11                             9  10 


2713     6     5i 

£    *.    d.  £.    s    d. 

4  Multiply  37  6  9i  by  5         Prod.  186  13  Hi 

5  —        56  8  7|  by  9          —    507  17     9| 

COMPOUND  DIVISION. 

COMPOUND  DIVISION  is  the  art  of  dividing  a  sum 
which  consists  of  several  denominations. 

CASE  1. 
When  the  divisor  does  not  exceed  12. 

RULE. 

Divide  the  several  denominations  of  the  given  sum, 
one  after  another,  beginning  with  the  highest,  and  set 
their  respective  quotients  underneath. 

When  a  remainder  occurs,  reduce  it  to  the  neXt  lower 
denomination,  and  add  it  to  the  number  of  ihe  next  de- 
nomination, and  divide  the  sum  as  before. 


COMPOUND    DIVISION.  81 


EXAMPLES, 

bu.    pe.     qt.    pt. 

7)  25        2        6        1  ^ere  ^  *nto  ^  ku.  ^  times  and 

' 4  remain.      Set  down  the  3. 

Reduce  the  4  bushels  to  pecks, 

3251        which  makes    16  pecks:    add  16 
pecks  to  2  pecks,  which  make  18  "peeks. 

Now  7  into  18  jpe.  2  times,  and  leave  4.     Set  down  the  2. 
Reduce  the  4  pecks  to  quarts,  which  makes  32  qts.     Add  32  qt.  to 
6  qt. — makes  38  qt. 

7  into  38  qt.  5  times,  and  3  remain.     Set  down  the  5. 
Reduce  the  3  qt.  to  pt. — makes   6  pt.,  add  6  pt.  to  1  pt.  makes  7 
pints. 

7  into  7,  1  time.     Set  down  the  1,  and  the  work  is  completed. 

bu.      pe.      qt.  bu.      pe.      qt. 

2)  8        2        6  3)  9        3        6 


4         1         3 

4  Divide  34  bu.  3  pe.  6  qt.  between  9  persons. 

Ans.  3  bu.  3  pe.  4  qt. 

5  92  bu.  3  pe.  belong  equally  to  7  persons;  what  is  the 
share  of  each?  Ans.  13  bu.  1  pe. 

CASE  2. 

When  the  divisor  exceeds  12,  and  is  the  exact  product 
of  two  numbers  in  the  multiplication  table. 

RULE. 

Divide  the  sum  by  one  of  the  factors,  and  the  quotient 
by  the  other. 

Multiply  the  last  remainder  by  the  first  divisor,  and 
add  the  first  remainder  for  the  true  remainder,  as  in 
simple  division,  note  2. 

EXAMPLES. 

1  Divide  89  bu.  3  pe.  7  qt.  by  28. 

Quotient  3  bu.  6  qt.  1  pt. — 18  .era. 


2 


82  COMPOUND    DIVISION. 


'4 


bu. 

89 


22     1     7—3  Rem, 


3     0     6 — 5  Rem. 

4  First  divisor 

20 

3  First  rem. 

True  rem.  23  Quarts 
2 

28)46  pints 

Pt.  1  and  a  rem,  of  18  pints  undivided. 


3.  Divide  78  bushels,  3  pecks,  4  quarts,  among  32 
persons;  what  will  be  the  share  of  each? 

Ans.  2  bu.  1  pe.  6  qt.  1  pt.  and  a  remainder  of  24 
pints  undivided. 


CASE  3. 

When  the  divisor  is  more  than  12,  and  is  NOT  the  exact 
product  of  any  two  numbers  in  the  multiplication  table 

RULE. 

Divide  the  highest  denomination  of  the  given  sum,  as 
in  case  2,  simple  division;  and  reduce  the  remainder, 
if  any,  to  the  next  lower  denomination;  add  the  number 
of  that  denomination  to  the  result,  and  divide  as  before. 


EXAMPLES. 
1.  Divide  77  bushels,  1  peck,  7  quarts,  by  23. 

Quotient,  3  bu.  1  pe.  3  qt.  1  pt.!3rem. 


COMPOUND    DIVISION.  83 

EXAMPLES. 

bu.  pe.  qt.bu.pe.qt.  pt. 
23)79     1     7(3     1     6     1+3  pint  remaining. 
69 

10 

4 

23)41(1  peck 
23 


18 
8 

23)151(6  quarts 
138 

13 

2 

23)20(1  pint 
23 

Rem.       3  pints 

2.  A  boat  load  of  corn,  containing  4927  bushels,  3 
pecks,  is  owned  equally  by  29  persons :  wnat  is  the 
share  of  each? 

Ans.  169  bu.  3  pe.  5  qt.  1  pt.,  and  a  rem.  of  1  pint. 

Questions. 

What  is  compound  division? 

When  the  divisor  does  not  exceed  12,  how  do  you 
proceed  ? 

When  a  remainder  occurs,  what  do  you  do  with  it? 

Where  the  divisor  exceeds  12,  but  is  the  product  of  two 
numbers  in  the  table,  how  do  you  perform  the  operation  ? 

How  do  you  find  the  true  remainder  in  the  latter  case  ? 

When  the  divisor  is  more  than  12,  and  is  not  the  pro- 
duel  of  any  two  numbers  in  the  table,  how  do  you  per- 
form the  operation? 


84  COMPOUND    DIVISION. 

AVOIRDUPOIS  WEIGHT. 
tons     cwt.      qr.       Ib.  Ib.          oz.         dr. 

6)37         17        3        27          7)40         12         14 

6619-1  rem.  6         10         15-5  R. 


tons      cwt.      qr.  cwt.       qr.         Ib.         oz.       dr. 

8)92        3        3  9)75        3        23         14         12 


5.  A   quantity  of  iron  weighing  473  tons,    19  cwt., 
3 quarters,  is  owned  equally  by  22  persons;  what  is  the 
share  of  each? 
Ans.  21  T.  10  cwt.  3  qr.  16  Ib.  8  oz.  11  dr.  Rem.  14  dr. 

APOTHECARIES  WEIGHT. 

fc     3     3    9  &     3    3    9    gr. 

4)23     7     5     1  5)41     6     7     2     14 


5  10    7     1  8361       2-4rem. 

fc     5     3    9  fc     3    3    9    ST. 

6)46     912  7)93     7     5    2     14 


5.  Divide  127fe   3g   63  into  17  equal  parcels:  how 
much  in  each  parcel?   Ans.  7fe  5g  63  2^  16gr.  8  rem. 

TROY  WEIGHT. 

Ib      oz.    dwt.  Ib.    oz.   dwt.  gr. 

8)34     10     15  7)45     11     16    22 


Ib.    oz.  dwt.  Ib.    oz.  dwt.  gr. 

9)78     9     16  8)82     7     14    21 


COMPOUND    DIVISION.  85 

CLOTH  MEASUEE. 
yd.       qr.      na.         Ells  E.   qr.     na. 
5)27        3         1          6)37        3 


o 


1         1+4  rem. 


yd.       qr.     na.          EllsE.  qr.      na. 
7)45        3        2          8)37        3         1 


LONG  MEASURE. 

1.          m.      f.  m.         f.        p.       yd. 

6)37        2         2  7)46        7        17        3 


607  6        5        25        2 

yd.       ft.      in.  m.        f.       p.       yd.    ft. 

6)53        2        9  7)87        6       23        4        2 


, 

5.  A  traveller  has  a  journey  of  946  miles,  6  furlongs, 
to  perform  in  26  days;  how  far  must,  he  travel  each 
day?  Ans.  36  m.  3  f.  12  p.  8  rem. 

LAND,  OR  SQUARE  MEASURE.      ; 

A.     R.  P.  A.    R.    P.  yds. 

7)37     3     27  9)423     3     28     2 

5     1     26+5  Rem.  47     0     16  13+5  Rem. 

3.  A  farm  containing  746  acres,  3  roods,  29  poles,  is 
to  be  divided  equally  between  9  heirs;  what  is  the  share 
of  each?  Ans.  82  A.  3  R.  38  P.  and  7  rem. 

CUBIC     MEASURE. 

cords       ft.  yd.       ft.          in. 

8)97        48  9)148       16        493 

12        22  16      13      1398+7R. 


86  COMPOUND    DIVISION. 

3.  A  boat  load  of  wood,  containing  92  cords  87  feet, 
is  to  be  divided  between  3  persons;    what  is  the  share 
of  each?  Ans.  30  c.  114  ft.    1  rem, 

4.  A  quantity  of  earth,  containing  6987  yards,  25 
feet,  is  to  be  removed  by  29  carters;   how  much  must 
each  remove?  Ans.  240  yd.  20  \ 

LIQUID  MEASURE. 
tuns.Jihd.  gal.  hhd.  gal.  qt.  pt. 

5)37     3    45  6)57     36     3     1 


7     2  .  21+3  Rem.         9    37    2     l-(-l  Rem 


tuns   hhd.  gal.  hhd.  gal.  qt.  pt. 

7)84    2     32  8)93    43     3     1 


5.  A  quantity  of  liquor  owned  equally  by  27  person.* 
the  whole  quantity  being  431  hhd.  47  gals;  what  i 
the  share  of  each?  Ans.  15  hhd.  62  gal.  1  qt.;  17  rem. 

MOTION. 

Sin.       °        '       "  Sin.     ° 

8)9     16    45     36  9)11     23    48     54 

1       5    50    42 


TIME. 

yr.         mo.  we.   da.  ho.  min.  sec 

11)848         10  12)24    6    20    32     24 


77          2  2    0     13    42    42 


yr.         mo.  da.  ho.    min.  s*,c. 

4)375        8  7)37     16    28     32 


COMPOUND   DIVISION.  87 

STERLING  MONEY. 

£    *.    d.  £    *.    d. 

6)82     14    6  8)143    7  10 


13     15     9  17  18     5| 

£      s.    d.  £      s.    d. 

7)78     10     11  9)98     17     1 


£       s.    d.£    s.     d. 
19)36     16    3(1     18    9 

6  Divide  113£  13s.  4d.  by  31.  What  is  the  quotient? 

Ans.  3£  135.  4d 

7  Divide  189£  14s.  by  95.      Quotient,  1£  19s 


PROMISCUOTIS    EXERCISES 

1  In  35  dollars  how  many  cents?  Ans.  3500. 

2  How  many  miles  are  there  in  98  furlongs  ? 

Ans.  12M.  2fur. 

3  How  many  weeks  are  there  in  365  days? 

Ans.  52we.  1  da. 

4  In  84  half  cents  how  many  cents?         Ans.  42cts. 

5  In  8  tons  15  cwt.,  how  many  hundred  weight? 

Ans.  175  cwt. 

6  How  many  perches  are  there  in  63  roods  ? 

Ans.  2520  square  per. 

7  How  many  pounds  in  157s.?  Ans.  £7  17s. 

8  In  175  pecks  how  many  bushels  ?      Ans.  43bu.  3pe. 

9  In  7642  cents  how  many  dollars  ?    Ans.  $76  42cts 

10  In  103  pints  how  many  quarts?         Ans.  51qt.  Ipt. 

11  How  many  minutes  are  there  in  720  seconds? 

Ans.  12min. 

12  In  7  hogsheads,  33  gallons,  how  many  gallons? 

Ans.  474  gal. 


88  SINGLE  RULE  OF  THREE, 

PROPORTION ; 

OR, 

THE  SINGLE  RULE  OF  THREE. 

PROPORTION  is  an  Equality  of  RATIOS  ;* 

That  is,  four  numbers  are  proportional,  when  the  first  has  the 
same  ratio  to  the  second  as  the  THIRD  has  to  the  FOURTH,  Thus> 
as  12  :  4  :  :  24  ;  8;  or  as  4  :  12  : :  8  :  24, 

The  ratio  of  12  to  4  is  3  and  the  ratio  of  24  to  8  is  3.  Or,  the 
ratio  of  4  to  12  is  £,  and  the  ratio  of  8  to  24  is  £.  Then 

Four  numbers  are  proportional,  when  the  first  is  as  many  times  the 
second  or  the  same  part  of  the  second,  as  the  third  is  of  the  fourth. 
Or,  when  the  ratio  of  the  first  to  the  second  equals  the  ratio  of  the 
third  to  the  fourth. 

Tha  two  quantities  compared  are  called  the  TERMS 
of  the  ratio:  the  first  is  called  the  ANTECEDENT,  and 
the  second  the  CONSEQUENT.  In  any  series  of  four 
proportionals,  the  first  and  fourth  terms  are  called  the 
EXTREMES,  and  the  second  and  third  the  MEANS.  The 
product  of  the  Means,  equals  the  product  of  the  Ex- 
tremes. Thus  in  either  series  above,  12X8=96,  and 
24X4=96. 

Now  suppose  we  have  the  three  first  terms  of  a  series  in  propor- 
tion, and  we  wish  to  find  the  fourth.  Divide  the  product  of  the 
second  and  third  terms  by  the  first,  and  the  quotient  will  be  the 
fourth  term.  In  this  manner  let  the  fourth  term  be  found  in  each 
of  the  following  series. 

2  :  4  :  :  8  is  to  what?  Am.  16. 

3  :  11  :  :  9  is  to  what?  tins.  33. 

4  :  6  :  :  6  is  to  what  ?  Jlns.  9. 

2  :  9  :  :  8  is  to  what?  Jlns.  30. 

5  :  7  :  :  15  is  to"  what?  Am.  21. 


RULE. 

number  which  is  of  the  name  or  kind  in 
which  the  answer  is  required,  in  th<?  third  place: 


Set  that 


*  RATIO  is  the  relation  of  one  thing  to  another  of  the  same  kind 
in  regard  to  magnitude  or  quantity. 


SINGLE  RULE  OF  THREE.  89 

And,  if  the  answer  must  be  greater  than  the  third 
term,  set  the  greater  of  the  remaining  two  terms  in  the 
second,  and  the  less  in  the  first  place  ;  but,  if  the  an- 
swer must  be  less  than  the  third  term,  set  the  less  in  the 
second,  and  the  greater  in  the  first  place. 

When  the  first  and  second  terms  are  not  of  the  same 
denomination,  reduce  one  or  both  of  them  titlt  they  are; 
and,  if  the  third  consist  of  several  denominations,  reduce 
it  to  the  lowest,  then 

Multiply  the  second  and  third  terms  together  and 
divide  the  product  by  the  FIRST,  and  the  quotient  will 
be  the  fourth  term  or  answer. 

NOTE.  —  The  answer  will  be  of  the  same  denomination  as  the 
third  term  ,•  and,  in  many  instances,  must  be  reduced  to  a  greater 
denomination. 

EXAMPLES. 

1.  If  four  pounds  of  sugar  cost  50  cents,  what  will 
24  pounds  cost  at  the  same  rate?  Ans.  $3,00. 

1st  Term.  2d  Term.  3d  Term. 

tLj_^_lu_j    t^^^-^j  y_,-  —  ,_j          In  this  question  the  answer  is 

jj                j,  required  to  be  money:  therefore 

•**•  C"'         money  (the  50  cts.)  must  be  in 

As  4  the   third  place.       Because   24 

50  pounds  will  cost  more  than   4 

_  pounds  —  the  greater    (24   Ibs.) 

4)1200  must  occupy  the  second  place: 

'  _  and  the  remaining  term  (4  Ibs.) 

~ 


2.  If  24  pounds,  cost  300  cents  (or  3  dollars  ;)  how- 
many  pounds  may  be  purchased  for  50  cents  at  the 
same  rate?  Ans.  4  Ibs. 

cts.  els.  Ibs.          In  this  question  the  answer  is  re- 

As  300     :     50     :  :     24      quired  to  be  in  pounds,-   therefore 
50  pound8  (th6  24)  must  be  in  the  third 
place, 

Because  50  cts.  will  purchase  less 

300)1200   than  300  cts.—  the  lens   (50   cts.) 

-   must  occupy  the  second  place  :  and 

4   the  remaining  term   (300  cts.)  the 

first. 


90  SINGLE  RULE  OF  THREE. 

3.  Bought  a  load  of  corn  containing  27  bushels,  3 
pecks,  at  50  cts.  per  bushel,  what  did  it  cost? 

Ans.  $13,87^. 
bit.  bu.    pe.          cts. 

1      :      27     3    :  :    50 

4  4 

Because  pecks  occur  in  the  second 

term,  the  first  and  second  are  reduced 
4          111  to  pecks. 

50 

4)5550 

$13,87^ 

4.  What  are  42  gallons  worth,  if  3  gallons  2  quarts 
cost  $1,20?  Ans.  $14,40. 

gals.  qts.        gals.  D.  cts. 

3     2     :     42    :  :     1,20 

44  As  quarts  occur  in  the  first  term, 

—              the  first  and  second  are  reduced  to 

14               168  quarts- 
120 

14)20160 

$14,40 

5.  If  8  bushels  2  pecks  cost  $4,25,  how  many  bush- 
els can  I  purchase  with  $38,25  ?       Ans.  76  bu.  2  pe. 

D.  cts.             D.  cts.  bu.  pe. 

4,25      :      38,25  :  :    8    2 

34  4             As  two  denominations  occur 
in  the  third  term,  it  is  reduced 

•*A  r^     t0  the  leSS  ;   hellCe  the  reSult  'S 

d4pe.  pecks>  whicil  must  be  re<luced 
11475  to  bushels. 

425)130050(306  pe. 
1275 

pe. 
2550   4)306 

2550    

76  bu.  2  pe. 


SINGLE  RULE  OF  THREE. 


91 


G.  What  will  5  Ib.  6  oz.  5  d\vt.  of  silver-ware  cost 
at  $1,50  per  ounce?  Ans.  $99,37^. 


oz. 
1 
20 

20 


U).  oz.  dwt. 
:      565 
12 

66 
20 

1325 
150 

66250 
1325 


D.  ds. 
1,50 


As  dwts.  arc  in  the  second 
term,  the  first  and  second 
must  be  reduced  to  dwts. 


. 


20)198750 

f99,37| 

7.  AVhen  3  yards  and  8  feet  of  plastering  cost  $1,40, 
what  will  be  the  cost  of  16  yards?  Ans.  $5,76. 

ydv.ff.  yds.  D.cts. 

38       :       16       :  :        1,40 
9  9 

35  144 

140 

5760 
144 

35)20160(5,76  cts. 

8.  How  many  yards  of  cloth  can  be  purchased  for  95 
collars,  if  4  yd.  3  qr.  cost  $9,50  ? 

Ans.  47  yd.  2  qr. ;  or  47|  yd. 

$  ct.        $  ct.  yd.  qr.        qr. 

As?  9,50  :  95,00    :  :    4     3      4)190(47  yd.  2  qr. 


92  SINGLE  RULE  OF  THREE. 

NOTE. — The  operation  may,  in  many  instances,  be  contracted  by 

dividing  the  second  or  third  term  by  the  first ;  or  the  first  by  either 

|!  of  the  others,  or  by  any  number  that  will  divide  the  first  and  either 

of  the  others  without  a  remainder;  and,  using  the  quotients  instead 

of  the  original  numbers. 

9.  If  24  yards  cost  $96,  what  will  8  yards  cost? 

Aus.  $32. 

yds.         yds.  D.  yds.      yds.  D. 

24a     :    *8a     :  :     96c    or    24a     :     8     :  :     96a 

3c  •     Ans.  32.  —  4 

32 
10  If  36  bushels  cost  $72;  what  will  12  bu.  cost? 

Ans.  $24. 

bu.  bu.  D.  bu.  bu.  D. 

3Ga  :  I2a  :  :  72c    12)36  :  12  :  :  72a 

3c  24        3a    1      24 

APPLICATION. 

1  When  4  bushels  of  apples  cost  $2,25,  what  must 
be  paid  for  20  bushels  ?  Ans.  $11,25. 

bit.        bu.          D.  cts. 
4     :     20     :  :     2,25 

2  How  many  yards  of  cloth  can  I  buy  for  $60,  when 
5  yards  cost  $12?  Ans.  25  yds. 

D.          D.         yds. 
12     :     60     :  :     5 

3  If  6  horses  eat  21  bushels  of  oats  in  a  given  time ; 
how  much  will  20  horses  eat  in  the  same  time  ? 

Ans.  70  bu. 

,4  If  20  horses  eat  70  bushels  of  oats  in  a  certain  time ; 
how  much  will  4  horses  eat  in  the  same  time? 

Ans.  14  bu. 

5  If  a  family  ofHen  persons  use  7  bushels  3  pecks  of 
wheat  in  a  month;  how  much  will  serve  them  when 
there  are  30  in  the  family?  Ans.  23  bu.  1  pe. 

6  If  14  Ibs.  of  sugar  cost  75  cents,  how  many  pounds 
can  be  bought  for  three  dollars?  Ans.  56  Ibs. 

7  If  4  hats  cost  12  dollars,  what  will  27  feats  cost  at 
the  same  rate?  Ans.  $81. 


SINGLE    RULE    OF    THREE.  93 

8  If  20  yards  of  cloth  cost  $85,  what  will  324  yards 
cost  at  the  same  rate  ? 

Ans.  $1377. 

9  If  2   gallons  of  molasses  cost  70  cents,  what  will  2 
hogsheads  cost  ?  Ans.  $44,10. 

10  If  1  yard  of  cloth  cost  $3,25  cts.,  what  will  be  the 
cost  of  6  pieces,  each  containing  12  yds.  2  qrs.  ? 

Ans.  $243,75  cts. 

11  If  3  paces  or  common  steps  of  a  person  be  equal  to 
2  yards,  how  many  yards  will  160  paces  make? 

Ans.   106  yds.  2  ft. 

12  If  a  person  can  count  300  in  2  minutes,  how  many 
can  he  count  in  a  day  ?  Ans.  216000. 

13  What  quantity  of  wine  at  60  cts.  per  gallon  can  be 
bought  for  $37,80  cts.  Ans.  63  gal. 

14  If  8  persons  drink  a  barrel  of  cider  in  10  days,  how 
many  persons  would  it  require  to  drink  a  barrel  in  4  days? 

Ans.  20. 

15  If  8  yards  of  cloth  cost  $12,  what  will  32  yards 
[cost?  Ans.  $48. 

16  If  3  bushels  of  corn  cost  $1,20,  what  will  13  bush- 
els cost?  Ans.  $5,20. 

17  If  9   dollars  will  buy  6  yards  of  cloth,  how  many 
yards  will  30  dollars  b^y  ?  Ans.  20. 

18  If  a  man  drink  3  gills  of  spirits  in  a  day,  how  much 
will  he  drink  in  a  year  ?  Ans.  34  gal.  1  pt.  3  gi. 

19  If  12  horses  eat  30  bushels  of  oats  in  a  week,  how 
many  bushels  will  serve  44  horses  the  same  time  ? 

-Ans.  110. 

20  If  a  perpendicular  staff  6  feet  long,  cast  a  shadow  5 
feet  4  inches,  how  high  is  that  tree  whose  shadow  is 
104  feet  long  at  the  same  time?  Ans.  117  feet. 

EXERCISES. 

1  If  12   acres,  2  roods,  produce  525  bushels  of  corn, 
how  many  bushels  will  62  acres,  2  roods  produce  ? 

Ans.  2625  bu. 

2  If  7  men  plough  6  acres,  3  roods  in  a  certain  time, 
how  many  acres  will  96  men  plough  in  the  same  time  ? 

Ans.  92  A.  2  R.  11  Per.  12  yd.-f 


94  SINGLE   RULE   OF   THREE. 

3  Suppose  3  men  lay  9  squares*  of  flooring  in  2  day  s ; 
low  many  men  must  be  employed  to  lay  45  squares  in 
he  same  time?  Ans.  15  men. 

4  If  7  pavers  lay  210  yards  of  pavement  in  one  day; 
low  many  pavers  would  be  required  to  lay  120  yards  in 
he  same  time?  Ans. 4  pavers. 

5  If  2  hands  saw  360  square  feet  of  oak  timber  in  2 
days;  how  many  feet  will  8  hands  saw  in  the  same  time? 

Ans.  1440  feet. 

6  An  engineer  having  raised  a  certain  work  one  hun- 
dred yards  in  24  days,  with  5  men;  how  many  men  must 
be  employed  to  perform  a  like  quantity  in  15  days? 

Ans.  8  men. 

7  If  3  paces  or  common  steps  be  equal  to  2  yards ;  how 
many  vards  will  160  such  paces  make? 

Ans.  108  yd.  2  ft. 

8  If  a  carriage  wheel  in  turning  twice  round,  advance 
33  feet  10  inches;  how  far  would  it  go  in  turning  round 
63360  times?  Ans.  203  miles. 

9  Sound  flies  at  the  rate  of  1142  feet  in  1  second  of 
time;  how  far  off  may  the  report  of  a  gun  be  heard  in  1 
minute  and  3  seconds? 

Ans.  13  miles,  5  fur.,  0  poles,  2  yd. 

10  If  a  carter  haul  100  bushelst>f  coal  at  every  3  loads; 
how  many  days  will  it  require  for  him  to  load  a  boat  with 
3600  bushels,*  suppose  he  haul  9  loads  a  day? 

Ans.  12  days. 

bu,        lu.        da.        da. 
As  300  :  3600  :  :    1     :     12. 

11  If  8  men  can  reap  a  field  of  wheat  in  4  days;  how 
many  days  will  it  require  for  16  men  to  do  it? 

Ans.  2  days. 

12  -Sold  10  yards  of  linen  at  5  dollars  50  cents;  what 
was  it  a  yard?  Ans.  55  cents. 


*A  SQUARE  is  10  feet  long  and  10  feet  wide,  or  100  square  feet. 
This  measure  is  employed  in  estimating  the  quantity  of  flooring,  roof- 
ing, weather-boarding,  &c. 


SINGLE    RULE    OF    THREE. 


95 


13  If  7  pounds  of  cheese  cost  87  i  cents;  what  must  I 
pay  for  122  pounds?  Ans.  15  dol.  25  ct. 

14  If  1  ounce  of  silver  cost  72  cents;  what  will  3  pounds 
5  ounces  come  to?  Ans.  29  dol .  52  ct. 


Why  do  you  multiply  the  second  and  third  terms  to- 
gether and  divide  by  the  first? 

What  will  24  pounds 
of  bacon  cost  at  50  ct. 
for  every  4  pounds? 
Ib.     Ib.    ct.     ct. 


As    4  :  24  ::50  : 300 


cts. 
4)50 

12}  the  price  of  lib. 
24 

50 
25 


If  4  pounds  cost  50  cents,  divide  the 
50  cents  by  4,  gives  the  price  of 
1  pound:  thus  4  into  50  12*.  If 
1  pound  cost  12£  cents,  2  pounds 
will  cost  twice  that;  three  pounds 
three  times,  and  24  pounds  will  cost 
24  times  12!  ct. ;  that  is  300  cts.  or  3 
dol. 


If  the  second  and  third  terms  be 
multiplied  together,  and  their  product 
divided  by  the  first,  the  result  will  be 
the  Bame  as  it  is  when  the  third  is  di- 
vided by  the  first,  and  the  quotient 
multiplied  by  the  second. 


300  ct.  the  price  of  24  Ib. 

15  If  15  yards  of  broad  cloth  cost  80  dollars;  what  will 
75  cost.  Ans.  400  dollars. 

16  A  man  bought  li  yards  linen  for  $2  50  cts.;  what 
is  the  worth  of  1  qr.  2  na.  at  the  same  rate. 

Ans.  62*  ct. 

17  If  321  bushels  of  salt  cost  $240  75  cents;  what 
was  it  per  bushel?  Ans.  75  cents. 

18  If  the  moon  move  13  deg.  10  min.  35  sec.  in  one 
day;  in  what  time  does  it  perform  one  revolution? 

Ans.  27  da.  7  hrs.  43  min. 

deg.  min.  sec.  deg.  da. 
As  13     10    35:360  ::1 


96  SINGLE    RULE    OF    THREE. 

19  If  a  staff  4  feet  long,  cast  a  shade  7  feet  on  level 
ground ;  how  high   is  a  steeple  whose  shade  is  at  the 
same  time  198  feet.  Ans.  H3\ 

20  If  a  man's  annual  income  be  1333  dollars,  and  he 
spend  2  dollars  14  cents  a  day ;  what  will  he  save  at  the 
end  of  one  year?  Ans.  $551  90  ct. 

21  Suppose  A.  owes  B.  791  dols.  60  ct.,  and  can  pay 
only  374  cts.  on  the  dollar;  how  much  must  B.  receive? 

Ans.  $296  85  ct. 

22  Bought  3  casks  of  raisins,  each  containing  3  cwt 
1  qr.  14  lb.;  how  much  did  they  cost  at  $6  21  ct.  pei 
cwt.?  Ans.  $62,874 

PROPORTION— DIRECT  AND  INVERSE. 

Hitherto,  proportion  has  been  treated  in  general  terms ; 
it  now  remains  to  consider  the  two  kinds,  DIRECT  and 
INVERSE. 

DIRECT  PROPORTION  is  that  in  which  more  requires 
more,  or  LESS  requires  LESS.  Thus: 

yd.       yd.       dol.  If  2  yd.  cost  4  dol.,  124  yd.  being 

As  2    *  I'M:  •  *  4  more  than  2  yd.,  will  cost  more  than 

4  dol. 

yd.       yd.     dol.  And,  if  124  yd.  cost  248  dol.;  2yd. 

As    124   •  2  •  •  248         being  leis  wil1  cost  less" 

That  is,  more  yards  require  more 
money,  and  less  yards  cost  less  money 

INVERSE  PROPORTION  is  that  in  which  more  requires 
less;  and  less  requires  more.  Thus: 

If  12  men  built  a  wall  It  is  supposed  that  12  men  perform 
in  4  davs;  how  many  ft.  PiecJ  of*ork  I"  *  dars:wa , like 

"'i     •    •     o  j         n     P18ce  °»  work  is  to  be  done  in  o  days; 

men  can  do  it  in  8  days  i   this  will  require  a  kss  uumnei  of  men : 
that  is.  more  days  require  less  men. 

STATED.  Here    it  is  supposed  that   12  men 

da.  dct.  m.   m.   performed  a  piece  of  work  in  8  days: 

As  4 : 8  inversely : :  12 .  6   »  nke  .£iece  Is  to  be  done  in  ?  d*y.s  J 

this  will  require  more  men.      1  hat  is, 
more  days  require  less  men,  and  less 
da.  da.  m.   m.   days  require  more  men. 

As 8:  4  directly::  12.6 


SINGLE   RULE   OP   THREE.  97 

All  the  past  exercises  in  proportion  are  Direct — the 
"ollowing  will  be 

INVERSE  PROPORTION. 

Questions  in  Inverse  Proportion,  may  be  stated  and 
solved  by  the  same  rule  that  is  given  for  Direct  Propor- 
tion. 


EXERCISES. 

1  If  6  mowers  mow  a  meadow  in  12  days;  in  what 
time  will  24  mowers  do  it?  Ans,  3  da. 

2  If  a  man   perform  a  journey  in  6  days,  when  the 
days  are  8  hours  long;  in  what  time  will  he  do  it  when 
they  are  12  hours  long?  Ans.  4  da. 

3  If,  when  wheat  is  83  cents  a  bushel,  the  cent  loaf 
weighs  9  ounces ;  what  ought  it  to  weigh  when  wheat  is 
$  1  244  cts.  a  bushel?  Ans.  6  oz. 

4  If  100  dollars  principal  in  12  months  gain  6  dollars 
interest;  what  principal  will  gain  the   same  interest  in 
8  months?  Ans.  $150. 

5.  If  12  inches  long  and  12  inches  wide,  make  1 
square  foot;  how  long  must  a  board  be  that  is  9  inches 
wide,  to  make  12  square  feet?  Ans.  16  ft. 

6  A.  lent  B.  500  dollars  for  6  months ;  how  long  must 
B.  lend  A.  220  dollars  to  be  equivalent? 

Ans.  13  months,  19  days.-f 

7  There  is  a  cistern  having  a  pipe  that  will  empty  it 
in  12  hours;  how  many  pipes  of  the  same  capacity  will 
empty  it  in  15  minutes  ?  Ans.  48  pipes. 

8  A  certain  building  was  raised  in  8  months  by  120 
workmen,  but  the  same  being  demolished,  it  is  required 
to  be  rebuilt  in  2  months;  how  many  workmen  must  be 
employed?  Ans.  480  men> 

9  If  for  48  dollars  225  cwt.  be  carried  512  miles;  how 
many  hundred  weight  may  be  carried  64  miles  for  the 
same  money?  Ans.  1800  cwt. 

*A  month  is  estimated  at  30  days,  unless  a  particular  month  C  . 
referred  to. 

_^_^____ 


98  SINGLE   RULE   OF   THREE. 

10  If  48  men  can  build  a  wall  in  24  days;  how  many 
men  can  do  it  in  192  days  ?  Ans.  6  men, 

11  How  many  yards  of  carpeting  2  ft.  6  in   breadth, 
will  cover  a  floor  that  is  27  feet  long  and  20  feet  wide  I 

Ans.  72. 

12  What  quantity  of  shalloon  that  is  3  quarters  wide, 
will  line  7i  yards  of  cloth  that  is  U  yd.  wide? 

Ans.  15  yd. 

13  How  many  yards  of  matting  that  is  3  quarters  wide, 
will  cover  a  floor  that  is  18  feet  wride  and  60  feet  long? 

Ans.  160. 

14  In  what  time  will  $600  gain  the  same  interest  that 
$80  will  gain  in  15  years?  Ans.  2  years. 

Questions* 

What  is  proportion? 
When  is  the  proportion  direct? 
When  is  it  inverse? 

Why  is  the  proportion  inverse  in.  the  last  question? 
A.  because  it  is  more  money  requiring  less  time. 

Why  is  the  proportion  inverse  in  the  llth  question? 
A.  because  the  shalloon  is  narrower  than  the  cloth; 
that  is  less  width  requiring  more  length. 

Why  is  the  10th  question  inverse? 

PROMISCUOUS   EXERCISES. 

1  A  certain  steeple  standing  upon  level  ground,  casts  a 
shadow  to  the  distance  of  633  feet  4  inches,   when  a 
staff  3  feet  long,  perpendicularly  erected,  casts  a  shadow 
of  6  feet  4  inches;  what  is  the  height  of  the  steeple? 

Ans.  300  ft. 

2  A  ship's  company  of  15  persons  is  supposed  to  have 
bread  enough  for  a  voyage,  allowing  each  person  8  oun- 
ces a  day,  when  they  take  up  a  crew  of  5  persons,  with 
whom  they  are  willing  to  share;  what  wi"  be  the  daily 
allowance  of  each  person  now?  Ans.  6  oz. 

3  Bought  215  yards  of  broad  cloth  at  6  dollars  a  yard; 


SINGLE   RULE   OF   THREE.  99 

what  was  the  prime  cost,  and  how  must  I  sell  it  per  yard 
to  gain  $135  on  the  whole. 

Aiis.  prime  cost  $1290,00;  to  be  sold  for  $G,62f  per 
yard. 

4  If  100  men  can  complete  a  piece  of  work  in  12  days; 
how  many  can  do  it  in  3  days?  Ans.  400  men. 

5  If  a  board  be  4|  inches  wide;  how  long  a  piece  will 
it  take  to  make  1  square  foot?  Ans.  32  in. 

G  A  pole,  whose  height  is  25  feet,  at  noon  casts  a 
shadow  to  the  distance  of  33  feet  10  inches ;  what  is  the 
breadth  of  a  river  which  runs  due  East  at  the  bottom  of 
a  tower  250  feet  high,  whose  shadow  extends  just  to  the 
opposite  edge  of  the  water?  Ans.  338  ft.  4  in. 

7  A  plain  of  a  certain  extent  having  supplied  a  body  of 
3000  horses  with  forage  for  18  days ;  how  long  would  it 
have  supplied  2000  horses  ?  Ans.  27  da. 

8  A  piece  of  ground   1  rod  wide  and   160  rods  long, 
makes  1  acre*;  how  wide  a  piece  must  I  have  across  the 
end  of  a  farm  32  rods  wide  to  make  an  acre  ? 

Ans.  5  rods. 

9  I  have  a  floor  24  feet  long,  and  15  feet  wide,  which  I 
wish  to  cover  with  carpeting  that  is  3  quarters  of  a  yard 
wide;  how  many  yards  must  I  buy. 

Ans.  53  yards,  1  foot. 

10  How  much  land  at  $2,50  an  acre  must  be  given  in 
exchange  for3GO  acres  worth  $3,75  an  acre? 

Ans.  540  acres. 

11  What  is  the  weight  of  a  pea  to  a  steel-yard,  which 
is  39  inches  from  the  centre  of  motion,  will  balance  a 
weight  of  208  Ibs.,  suspended  at  the  draught  end  3  quar- 
ters of  an  inch?  Ans.  4  Ib. 

12  If  $28  will  pay  for  the   carriage  of  6  cwt.   150 
miles ;  how  far  should  24  cwt.  be  carried  for  the  same 
money?  Ans.  37  i  miles. 


100  DOUBLE   RULE   OF   THREE. 

COMPOUND  PROPORTION; 

OR 

THE  DOUBLE  RULE  OF  THREE. 

DIRECT    AND   INVERSE. 

COMPOUND  PROPORTION  is  two  or  more  series  of  pro- 
portionals combined. 

Five,  seven,  nine,  or  other  odd  number  of  terms, 
is  always  given  to  find  a  sixth,  eighth,  or  tenth,  &c*,  or 
answer. 

Rule  for  the  Statement. 

Place  the  numbers  that  is  of  the  denomination  in  which 

the  answer  is  required  to  be,  in  the  third  place.     Then: 

Consider  separately  each   pair  of  similar  terms  and 

place  them  agreeably  to  the  rule  for  SIMPLE  PROPORTION. 

•An 
OR, 

Work  by  two  separate  statements  in  simple  propor- 
tion.* 

Rule  for  the  Solution. 

Reduce  the  several  pairs  of  terms  to  similar  denom- 
inations as  in  single  proportion,  and  the  last  to  the  lowest 
denomination  given :  Then 

Multiply  the  two  initials,  or  left  hand  terms  togethei 
for  a  DIVISOR,  and  the  other  three  for  a  DIVIDEND. 

Divide  the  latter  by  thejformer,  and  the  quotient  wil< 
be  the  answer,  in  that  denomination  to  which  the  third 
term  was  reduced. 

EXAMPLES. 

1.  If  6  men  in  8  days  build  40  rods  of  wall,  how  rnucl 
will  18  men  build  in  20  days?  Aiis.  300  rods 


*  It  would  be  well  for  the  pupil  to  work  each  sum  both  ways. 


DOUBLE   RULE   OP   THREE.  101 

The  answer  is  required  to  be 
rods       given  in  rods :  then  rods  must  be 
Afi          the  third  term.     If  6  men  build 
40  rods,  18  men  will  build  more,; 
then  more  (18  men)  must  occupy 
the  second,  and  less  (6  men)  the 

48  360  first  Place- 

If  8  days  produce  40  rods,  20 
days  will  produce  more;    then 
more  (20  days)  must  occupy  the 
48)14400(300  *eomd,  and  few  (8  days)  the>j* 

144  place. 

N.  B.  The   first  pair,  or  two 

upper  terms  must  be  alike.     Also 

the  lower  pair  must  be  alike. — 

That  is,  both  must  be  men  or  both  days,  both  hours  or  both  bushels,  &c. 

2.  If  6  men   in  eight  days  eat  lOlb.  of  bread,  how 
much  will  12  men  eat  in  24  days?  Ans  60. 

men   6   :  12)         in  IK 

days  8   :  24»    :  Contracted. 

— ^  6   :  12  2j 

288  8  :  24  3     ' 

10  — 
6 

48)2880(60  Ans.  10 

288  — 

60  Ans. 

0 

3.  Suppose  4  men   in  12  days  mow  48  acres,  how 
many  acres  can  8  men  mow  in  16  days?       Ans.  128A. 

4."  If  10  bushels  of  oats  be  sufficient  for  18  horses  20 
days,  how  many  bushels  will  serve  60  horses  36  days, 
at  that  rate?  Ans.  60bu. 

5.  Suppose  the  wages  of  six  persons  for  21  weeks  be 
288  dollars,  what  must  14  persons  receive  for  46  weeks? 

Ans.  $1472. 

6.  If  the  carriage  of  8cwt.  128  miles  cost  f  12.80, 
what  must  be  paid  for  the  carriage  of  4cwt.  32  miles? 

Ans.  $1.60. 

7.  If  371b.  of  beef  be  sufficient  for  12  persons  4  days, 
how  many  Ib.  will  suffice  38  men  16  days? 

Ans.  4681b.  lOi  oz. 


102  DOUBLE   RULE    OF    THREE. 

8.  If  a  man  can  travel  305  miles  m  30  days,  when 
the  days  are  14  hours  long,  in  how  many  days  can  he 
travel  1056*  miles,  when  the  days  are  12^  hours  long? 

Ans.  116days.-f2540. 

9.  If  the  carriage  of  24cwt.  for  45  miles  be  18  dol- 
lars, how  much  will  it  cost  to  convey  76cwt.  121  miles? 

Ans/$153  26  cts.+720. 

10.  A  person  having  engaged  to  remove  £000cwt.  in 
9  days;  removed  4500cwt.  in  6  days,  with   18  horses: 
how  many  horses  will  be  required  to  remove  the  balance 
in  the  remaining  3  days?  Ans.  28  horses. 

11.  If  3  men  reap  12  acres  3  roods  in  4  days  3  hours, 
how  many  acres  can  9  men  reap  in  17  days? 

Ans.  153  acres. 


Analysis. 

If  3  m.  reap  12  a.  3  r. 
1  m.  rea  •  4  a.  1  r.  and 
9  m.  reap  38  a.  1  r. 

If  4  d.  3h.  reap  38  a. 
1  r.  Id.  reap  9  a.  ami 
17  d.  reap  153  a.  Ans. 


1836 
9180 

153)93636(612  roods,  or  153  acrea 
918 


183 
153 

306 
306 


*The  day  is  here  estimated  at  twelve  hours. 


PRACTICE.  103 

12.  If  40  men  build  32  rods  of  wall  in  8  days,  work- 
ing 10  hours  each  day,-  in  how  many  days  will  60  men 
build  48  rods,  working  12  hours  a  day  ? 

Ans.  6  days,  8  hours. 

Men     men  Multiply   all   the   initial 

60  :     40  terms   (or  60,  32,  and  12) 

rods     rods  days  together  for  a  divisor:  and 

32   :     48          :   :  8     the  other   four  for  a  dim- 

hours    hours  dend. 

12   :     10 

,.  13.  If  36  men  dig  a  cellar  60  feet  long,  24  feet  wide, 
and  8  feet  deep,  in  16  days,  working  16  hours  per  day, 
ho\v  many  men  can  dig  a  cellar  80  feet  long,  40  feet 
wide,  and  12  feet  deep,  in  20  days,  working  12  hours  per 
day?  Ans.  128. 

Questions. 

What  is  ccfnpound  proportion? 

How  do  you  state  questions  in  compound  proportion? 
Which  terms  do  you  multiply  together  for  a  divisor? 
Which  for  a  dividend? 
What  other  method  is  there? 


PRACTICE. 

PRACTICE  is  a  short  and  expeditious  method  of  per- 
forming various  calculations  in  business. 

CASE  1. 

When  the  given  price  is  LESS  than  one  dollar. 
RULE. — Set  down   the  given  number  as  one  dollar, 
and  take  such  aliquot  part*  or  parts  of  that  number,  as 
the  price  is  of  one  dollar,  for  the  answer. 


*An  aliquot  part  of  a  number  is  any  number  that  will  divide  A 
without  a  remainder;  thus  4  is  an  aliquot  part  of  20;  and  8  of  40: 
and  25  cents  is  an  aliquot  part  of  100  cents,  (or  $1.)  because  25  c« 
are  contained  in  100  cts-,  an  even  number  of  times,  without  a  remain- 
der. 


104 

PRACTICE. 

TABLE  OF  ALIQUOT 

PARTS. 

CTS. 

CWT. 

ROODS. 

50 

1  -v 

1 

10            r 

roods 

33i 

2              i 

0 

""3 

o 

11 

>••» 

25 

4 

J 

4 

P 

20 

r 
•f 

»-*j 

2               iV 

p 

P 

121 
10 

o 
? 

p 

1    D- 

qr. 

0 

perches 
20               i 

o 

3 

V 

iff 

p" 

2                 i 

10             ,\] 

5* 

1 

TO 

1 

o 

20               |" 

o 

4. 

1 

2J 

Ibs. 

10               ^ 

p 

2 

I 

5T- 

16                 i 

o 

8               i 

14                 t- 

i 

5               i 

<  o 

8 

r* 

4 

^ 

•7                       l 
7                          T6" 

2            it 

EXAMPLES 

. 

1.  What  will  826  bushels  of  wheat  come  to  at  25 

cts. 

a  bushel? 

Ans.  $206  50  cts. 

cts. 

$ 

25     3 

r    |8 

26                   82(5  bushels,  at  one  dollar  a  bushel,  will 
cost  826  dollars  :    at  25  cents,  or  ^  of  a 

dh    1       kc  rcn         dollar,  it  will  cost  one  fourth  as  much. 
$    J  <6UO,OU 

2.  What  will  934  gallons  of  molasses  cost,  at  50  cts. 
a  gallon?                                                         Ans.  $467. 

cts. 

$ 

50    \ 

r      934 

$    467 

3.  What  will  1832  bushels  of  salt  cost,  at  75  cents  a 

bushel? 

Ans.  $1374. 

cts. 

<fc                                   At  50  cents  the  cost  will  be 

50     \ 

_  _ 

sP 
r      1832 

i  as  much  as  —  at  one  dollar. 
At  25  cents  the  cost  will  be 

25     ] 

r      - 

i  as  much  as  —  at  50  cts. 

916 

cost  at  50  c. 

458 

cost  at  25  c. 

$1374 

cost  at  75  c. 

50 
25 


PRACTICE.  105 

<J»  As  before ;  the  cost  at  50  cts.  will 

1H^2  ke  i  as  much  as — at  1  dollar. 

At  25  cts.  it  will  be  i  as  much 


916  ct.  at  50. 
458  ct.  at  25. 


as — at  1  dollar. 


$1374  ct.  at  75. 

4.  What  will  680  pounds  of  sugar  cost,  at  1 0  cents  a 
pound?  $68. 

5.  What  will  742  pounds  of  pork  cost,  at  61-  cts.  a 
pound?  Ans.  $46  37* cts. 

6.  What  must  I  pay  for  371  pounds  of  bacon,  at  12|- 
cts.  a  pound?  Ans.  $46  37i  cts. 

7.  How  much  will  8750  bushels  of  rye  cost,  at  62| 
cts.  a  bushel?  Ans.  $5468  75 cts. 

8.  How  much  must  be  paid  for  4360  square  feet  of 
marble,  at  87  i  cents  a  foot?  Ans.  $3815. 

9.  What  will  468  square  yards  of  plastering  cost,  at 
18|  cents  a  yard?  Ans.  $87  75  cts. 

10.  How  much  will  be  the  cost  of  laying  856  perches 
of  stone,  at  93|  cents  a  perch?  Ans.  $802  50  cts. 

11.  What  will  the  digging  of  a  cellar,  containing  180 
cubic  yards,  cost,  at  20  cents  a  yard  ?  Ans.  $36. 

12.  What  will  be  the  cost  of  hauling  248  cords  of 
wood,  at  3H  cents  a  cord?  Ans.  $77  50  cts. 

13.  What  must  be  paid  for  432  perches  of  stone,  at 
37i  cts.  a  perch  ?  Ans.  $162. 

14.  How  much  must  be  given  for  724  days  labor,  at 
56*  cents  a  day?  Ans.  $407  25. 

15.  What  will  742  bushels  cost  at  10  cts.      Ans.  $71  20 

16.  732  15  10980 

17.  732  20  14640 

18.  475  25  11875 

19.  684  30  20520 

20.  756  35  26460 

21.  927  40  37080 

22.  824  50  41200 

23.  682  55  375  10 

24.  341  60  20460 

25.  784  70  54880 

E  2 '~ 


106 

PRACTICE. 

28.  What  will   352  busnels  cost  at  64  cts.?  Ans.$22  00. 

29. 

436                         124                      54  50 

30. 

724                         18|                    135  75 

31. 

956                        314                    298  75 

32. 

742                        37*                    278  25 

33. 

274                        43|                   119  87* 

34. 

732                        56*                    411  75 

35. 

845                        624                  528  124 

36. 

684                        68|                    470  25 

37. 

274                        814                  222  624 

38. 

386                        93|                   361  874 

CASE  2. 

When  the  given  price  is  MORE  than  one  dollar. 

RULE.  —  Multiply   the  given  sum  by  the  number  of 

dollars, 

and  take  the  aliquot  part  or  parts  for  the  cents. 

!  as  in  Case  1. 

EXAMPLES. 

1.  What  will  342  cords  of  wood  cost,  at  3  dollars  75 

cents  a 

cord?                                            Ans.  $1282  50. 

cts. 

$ 

50 

r        342              342  cords  at  $1,  will  cost  $342;  at  $3 

3          it  will  cost  3  times  $342;  at  50  cts.  it 

will  cost  i  as  much  as  it  will  at  $1.  ;  and  at 

25  cents,  4  as  much  as  it  will  at  50  cts.  ; 

1026          which  added  together,  will  be   the  cost  at 

25     J 

r        171          $375. 

85  50 

1282  50 

2.  What  will   250  acr.  cost  at  $4  624      AJIS.  $1  156  25 

3. 

435                    5  874                 2555  624 

4. 

273                    6  124                 1672  124 

5. 

942                    7  374                 6947  25 

6. 

846                    368|                 3119624 

7. 

957                    5  75                  5502  75 

8. 

236                    6  93|                 1637  25 

9. 

754                    3  564                 2686  124 

10. 

932                  27  25                25397  00 

PRACTICE. 
CASE  3. 


107 


When   the   given  quantity  consists   of  several  denom- 
inations. 

RULE. — Multiply  the  given  price  by  the  number  of 
hundred  weight,  acres,  yards, or  pounds,  &/c.  and  take  the 
aliquot  parts  for  the  quarters,  roods,  feet,  or  ounces,  &c. 

EXAMPLES. 

1.  What  will  240  acres,  2  roods,  10  perches,  cost  at 
$  15  25  cents  an  acre  ?  Ans.  $3668  57*  cts. 


2  r. 


10  p. 


1525 
240 

61000 
3050 
762i 
951-J-2  rem. 

3668574 


2.  What  will  29  yards,  4  feet,  of  stone  pavement  cost, 


at  $2  25  cents  a  yard? 
3  square  feet 


Ans.  $66  25  cts. 


29 

2025 
450 
,75 
25 

6625 


108 


PRACTICE. 


3.  What   will   32  pounds   8   ounces  of  silver  cost, 
at  $15,62i  a  pound?  Ans.  $510  41  i. 


6  oz.  Troy 


2oz. 


$1562i 
32 

16 

3124 
4686 
78H 
26CH+2Rem. 

510  4H 


4.  What  will   27  cwt.  3  qrs.  cost,  at  $23  50  cts.  a 
cwt.?  Ans.  $652  12i. 

5.  What   will  be  the  cost  of  47  Ib.  10  oz.  (Troy)  at 
$1  25  cts.  Ans.  $59  79. 

6.  What  will  64  yds.  3  qrs.  cost,  at  $2  25  a  yard? 

Ans.  $145  68| . 

7    Sold  83  yards  2  qrs.  of  cloth  at  $10  50  a  yard; 
what  does  it  amount  to?  Ans.  $876  75. 

8.  What  will  the  laying  of  28  squares,  75  feet  of  floor- 
ing cost,  at  $2  25  cts.  a  square?  Ans.  $64  68|. 

9.  What  is  the  cost  of  27  cords,  96  feet  of  fire  wood,  at 
$3  75  a  cord.  Ans.  $104  064 

10.  What  is  the  value  of  428  gals.  3  qts.  at  $1  40  cts. 
a  gallon  ?  Ans.  $600  25  cts. 

11.  What  is  the  value  of  765  gals.  3  qt.  1  pt.  at  $2 18| 
cents  a  gallon?  Ans.  $1675  34|  cts. 

12.  What  is  the  value  of  5  hhds.  3H  gals,  at  $47  12 
cts.  a  hogshead?  Ans.  $259  16  cts. 

13.  What  is  the  value  of  17  hhds.  15  gals.  3  qts.  at 
$64  75  cts.  per  hogshead?          Ans.  $1116  93  cts.  7m. 

14.  What  is  the  value  of  120  bu.  2  pecks,  at  35  cents 
a  bushel?  Ans.  $42  17  cts.  5  m. 

15.  What  is  the  value  of  780  bu.  2  pecks,  2  qts.  at  $1 
17  cts.  a  bushel?  Ans.  $913  25  cts.+ 

16.  What  is  the  value  of  1354  bu.  1  peck,  5  qts.  1  pt. 
at  25  cts.  a  bushel?  Ans.   $338  60  cts.  5m.+ 

17.  What  is  the  value  of  35  acres  2  roods  18  perches, 
at  51  dollars  35  cts.  an  acre?     Ans.  $1935  53  cts.  9m. 


TARE   AND   TEET.  109 

Questions 

What  is  practice? 

What  is  the  rule  for  the  solution  of  questions  in  prac- 
tice? 

What  is  an  aliquot  part? 

Are  50  cts.  an  aliquot  part  of  100  cents? 

What  part  of  $1  is  fifty  cents? 

What  part  of  $1  is  33i  cents? 

What  part  of  $1  is  25  cents? 

What  part  of  $1  is  12i  cents? 

What  part  of  $1  is  10  cents? 

What  part  of  $1  is  20  cents? 

What  part  of  $1  is  5  cents? 

What  part  of  $1  is  4  cents? 

What  part  of  $1  is  6*  cents? 


TARE  AND  TRET. 

TARE  AND  TRET  are  allowances  made  on  the  weight 
of  some  particular  commodities. 

Gross  weight  is  the  weight  of  the  goods,  together  with 
the  vessel  that  contains  them. 

Tare  is  an  allowance  for  the  weight  of  the  vessel. 

Tret*  is  an  allowance  of  4  Ib.  for  every  104,  for 
waste,  &,c. 

Neat  weight  is  the  weight  of  the  goods,  after  all  allow- 
ances are  made. 

BULB. 

Subtract  the  tare  from  the  gross,  and  the  remainder 
is  the  neat  weight. 

EXAMPLES. 

1  Bought  a  chest  of  tea,  weighing  gross  63  Ib.,  tare  8 
Ib. — what  are  the  neat  weight  and  value,  at  85  cents 
per  Ib? 

*  As  tret  is  never  regularly  allowed  in  this  country ;  no  account  -of  it 
is  taken  in  this  work. 

To 


110  TARE   AND    TRET. 

lb.  85  ct. 

63  gross — or,  weight  of  the  chest  and  tea     55  lb. 

8  tare— or,  weight  of  the  chest 
—  425 

55  neat — or,  weight  of  the  tea  itself       425 

$40,75  value, 

2  Bought  5  bags  of  coffee,  weighing  each  97  lb.  gross, 
tare  of  the  whole  7  lb. — what  are  the  neat  weight  arid 
value,  at  25  cents  per  lb.?  Ans.  478  lb.  neat— $119,50. 

3  The  gross  weight  of  a  hogshead  of  sulphur  is  1344 
lb.;  the  tare  138  lb. — what  are  the  neat  weight  and  its 
value,  at  $4,75  per  100  lb.? 

Ans.  1206  lb.  neat— $57,28i 

4  Bought  3  barrels  of  sugar,  weighing  as  follows,  viz 
236  lb.  gross,  23  lb.  tare— 217  lb.  gross,  22  lb.  tare— 
^25  lb.  gross,  23  lb.  tare — what  are  the  neat  weight  and 
value,  at  $8  per  100  lb.?      Ans.  610  lb.  neat— $48,80. 

5  Sold  3  hogsheads  of  sugar,  weighing  each  12  cwt.  2 
qrs.  14  lb.  gross;  tare  2  cwt.  1  qr.  27  lb. — what  are  the 
neat  weight  and  value,  at  $11,50  per  cwt.? 

Ans.  35  cwt.  1  qr.  15  lb.  neat — $406  91  i  cts. 

6  What  is  the  neat  weight  of  15  tierces  of  rice,  weigh- 
ing 48  cwt.  3  qrs.  12  lb.  gross;  tare  6  cwt.  12  lb.,  and 
what  is  the  value,  at  $5,25  per  cwt.  ? 

Ans.  42  cwt.  3  qrs.  neat— $224,43$. 

7  What  is  the  neat  weight  of  28  hogsheads  of  tobacco, 
weighing  201  cwt.  3  qrs.   12  ib.  gross;   tare  3140  lb.; 
and  what  does  it  come  to  at  $5  per  cwt.  ? 

Ans.  173  cwt.  3  qrs.  8  lb— $869  10|  cts. 

8  Bought  17  bags  of  grain,  weighing  3561  lb.  gross; 
tare  2  lb.  per  bag — what  is  the  neat?         Ans.  3527  lb. 

9  What  is  the  neat  weight  of  16  bags  of  pepper,  each 
weighing  65  lb.  gross;  tare  1ft  lb.  per  bag — and  what  is 
the  amount  at  30  cents  per  lb.? 

Ans.  1016  lb.  neat— $304,80. 


TARE   AND    TRET.  Ill 

10  In  14  hogsheads  of  sugar,  weighing  89  cwt.  3  qrs.i 
17  Ib.  gross;   tare  100   Ib.  per  hhd. — how  much   ne&t 
weight, and  what  is  its  value,  at  $9  per  cwt.? 

Ans.  77  cwt.  1  qr.  17  Ib.  neat— $696,611. 

11  What  are  the  neat  weight  and  value  of  16  hhds.  of 
tobacco,  each   weighing  5  cwt.   1  qr.  19  Ib.  gross;  tare 
101  Ib.  per  hhd.,  at  £2    6s.  lOd.  per  cwt.? 

Ans.  72  cwt.  1  qr.  4  Ib.  neat— £169.  5s.  4id. 

12  Bought  6  hhds.  of  sugar,  each  1126  Ib." gross;  tare 
117  Ib.  per  hhd. — what  are  the  neat  weight  and  value  at 
$8,75  per  cwt.?  Ans.  6054  Ibs.  $529,72*. 

13  What  are  the  neat  weight  and  cost  of  a  hogshead 
of  sugar  weighing  gross  986  Ib. ;  tare  12  per  cent,  (or 
12  Ib.  for  every  WO  Ib.)  at  $8  per  neat  hundred  pounds? 

Ib.     Ib.       Ib.       Ib.  Ib.     Ib.       Ib.       Ib. 


AslOO:986::  12:118, 

Ib. 

986         gross. 
118,        tare. 

868,        neat  weight. 


Or  as  100:  88::  986:  868, 


Ib. 

883 


8  dol. 


$69,44  the  value. 


14  What  are  the  neat  weight  and  value  of  4  hhds.  of 
sugar  weighing  gross  45001b.  tare  12  Ib.  per  cent,  at  $8, 
75  percent.?  Ans.  3960  Ibs.  neat        —$346,50. 

15  Bought  10  hhds.  of  sugar,  each  920  Ib.  gross;  tare 
10  Ib.  percent. — what  arelhe  neat  weight  and  value 
at  $-9,25  per  cwt.?  Ans.  8280  Ib.  neat-  $765,90. 

16  Sold  3  casks  of  alum,  each  675  Ib.  gross;   tare  13 
Ib.  percent. — what  are  the  neat  weight  and  value  at  $4, 
25  per  cent.  Ans.  1762  Ib.  neat— $74,87.4375. 

Or,  1762   Ib.  neat— $74,88. 5nearly. 

17  What  is  the  neat  weight  of48001b.gross:  tare  12  Ib. 
per  cent.?  Ans.  4224  Ib. 

18  What  are  the  neat  weight  and  value  of  4  hhds.  of 
sugar,  each  12  cwt.  1  qr.  14  Ib.  gross;   tare  12  Ib.  per 
cwt.  at  $12,20  per  cwt.? 

Ans.  44  cwt.  22  Il>  neat— $539  19i  cts. 


112  INTEREST. 

19  Bought  17  hhds.  of  sugar,  weighing  201  cwt.  2  qrs. 
13  Ib.  gross;  tare  10  Ib.  per  cwt. — what  are  the  neat 
weight  and  value  at  $14  per  cwt.? 

Ans.  183  cwt.  2  qrs.  13  Ib.  neat-  $2570  62  i  cts. 

INTEREST. 

INTEREST  is  an  allowance  made  for  the  use  of  money. 

Principal  is  the  sum  for  which  interest  is  to  be  com- 
puted. 

Rate  per  cent,  per  annum  is  the  interest  of  100  dol- 
lars for  one  year. 

Amount  is  the  principal  and  interest  added  together. 

CASE  1. 
When  the  time  is  one  year  and  the  rate  per  cent,  is  any 

number  of  dollars. 

RULE. — Multiply  the  principal  by  the  rate  per  cent., 
and  divide  by  100  j  the  quotient  will  be  the  interest  for 
one  year. 

EXAMPLES. 

1.  What  is  the  interest  of  500  dollars  for  1  year,  at  6 
per  cent,  per  annum? 

$500 
6 

100-f-$30100  Ans. 

2.  What  is  the  interest  of  225  dollars  for  1  year,  at  Y 
dollars  per  cent,  per  annum?  Ans.  $15  75. 

3.  What  is  the  interest  of  384  dollars  50  cents,  for  1 
year,  at  5  dollars  per  cent,  per  annum?    Ans.  $19  22i. 

4.  What  is  the  amount  of  $275  for  1  year,  at  6  per 
cent,  per  annum?  Ans-  $291  50. 

$275 
6 

16,50  interest 
275,00  principal 

$291,50  amount 


i:\-TEKEST.  113 

5.  What  is  the  interest  of  1654  dollars  81  cents  for  1 
year,  at  5  dollars  per  cent,  per  annum?   Ans.  $82  74-)-. 

6.  What  is  the  interest  of  1500  dollars,  for  1  year,  at 
i  dollar  per  cent,  per  annum?  Ans.  $7  50. 

7.  What  is  the  amount  of  $736,  at  6  per  cent,  per 
annum,  for  1  year.  780  dols.  16. 

8.  What  is  the  interest  of  524  dollars,  for  1  year,  at 
5J  dollars  per  cent,  per  annum?  Ans.  $27  51. 

9.  What  would  be  the  interest  of  842  dollars,  for  1 
year,  at  54  dollars  per  cent,  per  annum?      Ans.  $46  31. 

CASE  2. 
When  the  interest  is  required  for  several  years. 

RULE. — Find  the  interest  for  one  year,  and  multiply 
the  interest  for  one  year  by  the  number  of  years. 
EXAMPLES. 

1.  What  is  the  interest  of  500  dollars,  for  4  years,  at 
6  dollars  per  cent,  per  annum? 

$500 
6 

<  3000 

4 

$12000   Ans, 

2.  What  will  be  the  interest  of  540  dollars,  for  2  years, 
at  5  dollars  per  cent,  per  annum?*  Ans.  $54  00. 

3.  What  would  be  the  interest  of  482  dollars,  for  7 
years,  at  6  dollars  per  cent,  per  annum?  Ans.  $202  44. 

4  What  is  the  amount  of  $736  81i  with  7  years,  nine 
months  interest  due  on  it,  at  6  per  cent,  per  annum? 

Ans.  $1079  43|. 

Note. — If  the  interest  is  required  for  years  and  months, 
multiply  the  interest  for  1  year  by  the  number  of  years, 
and  take  the  aliquot  parts  of  the  interest  for  1  year,  for 
the  months. 


1 


10* 


114 


Omo. 


3  mo. 


INTEREST. 

$736  814 
6 

4420,874  interest  1  year 

7 

30946,12i  interest  7  years 
2210,431  interest  6  months 
1105,21!-)-  3 


34261,78  interest  for  7  yr.  9  mo. 
73681,25  principal 

-$107943  03  amount 


5.  What  is  the  amount  of  $362  25  for  4  years  6  mo. 
at  6  per  cent,  per  annum?  Ans.  $460  ()5|, 

CASE  3v 
When  the  interest  is  required  for  any  number  of  months, 

weeks  or  days,  less  or  more  than  one  year. 
RULE. — Find  the  interest  of  the  given  sum  for  one 
year     Then,  by  proportion, 
As  1  year 

Is  to  the  given  time, 

So  is  the  interest  of  the  given  sum  (for  1  year) 
To  the  interest  for  the  time  required. 
Or  take  the  aliquot  parts  of  the  interest  for  one  year, 
for  the  given  time,  as  in  note,  Case  2. 

EXAMPLES. 

1.  What  is  the  interest  of  $560  for  2  years  and  6  mo. 
at  5  per  ct.  per  annum?  Ans    $70 

560 
5 


6  mo. 


2800  interest  for  1  year 
2  years 

5600 
1400 

$70  00  interest  for  2  years  6  months. 


INTEREST.  115 

2.  What  is  the  interest  of  325  dollars,  for  4  years  and 

2  months,  at  4  dollars  per  cent,  per  annum? 

Ans.  $54  16  cts.  6m. 

3.  What  is  the  interest  of  840  dollars  for  5  years  and 

3  months,  at  4  dollars  per  cent,  per  annum? 

Ans.  $176  40. 

4.  WTiat  is  the  interest  of  840  dollars,  for  5  years  and 

4  months,  at  7  dollars  per  cent,  per  annum? 

Ans.  $313  60. 

5.  What  is  the  interest  of  5GO  dollars,  for  4  months,  at 
6  dollars  per  cent,  per  annum? 

560 
6 

m.     m.      $  cts.     $  cts. 
100)33  60         As  12  :  4  : :  33  60  :  11  20  Ans. 

6.  What  is  the  interest  of  1200  dollars,  for  15  weeks, 
at  5 dollars  per  cent,  per  annum?  Ans.  $17  30. 

7.  What  will  be  the  interest  of  240  dollars,   for  61 
days,  at  4|  dollars  per  cent,  per  annum?  Ans.  $1  90.-J- 

8.  What  is  the  interest 'of  $1000,  for  14  months,  at  7 
per  cent,  per  annum?  Ans.  $81  60S. 

9.  What  is  the  interest  of  450  dollars,  for  6  months 
and  20  davs,  at  5i  dollars  per  cent,  per  annum? 

Ans.  $1375. 

10.  What  is  the  interest  of  375  dollars  25  cents,  for  3 
years  2  months  3  weeks  and  5  days,  at  6  dollars  per  ct. 
per  annum?  Ans.  $72  92.-J- 

11.  What  is  the  amount  of  $736  for  28  weeks,  at  10 
per  cent,  per  annum?  Ans.  $775  63. 

CASE  4. 

To  fad  the  interest  of  any  sum  for  any  number  of  days^ 
as  computed  at  banks. 

RULE. — Multiply  the  dollars  by  the  number  of  days, 
and  divide  by  6;  the  quotient  will  be  the  answer  in  mills. 

The  interest  of  any  number  of  dollars  for  60  days,  at 
6  per  cent,  will  be  exactly  the  number  of  cents;  and  if 
any  other  rate  per  cent,  is  required,  take  aliquot  parts, 
and  add  or  subtract  according  as  the  rate  per  cent,  is 
more  or  less  than  6. 


116  INTEREST. 

EXAMPLES. 

1.  What  is  the  interest  of  563  dollars,  for  60  days,  at 
6  per  cent,  per  annum — and  likewise  at  7  per  ct.  per  an.? 

Ans.  $5,63  at  6  per  cent. 
60  $6,56.8  at  7  per  cent. 


6)33780 

in.  at)  

6per[       5630  mills, 
cent.) 


$5630 
938 


Interest  at  7  per  cent.  6568  mills. 

2.  What  is  the  interest  of  854  dollars,  for  30  days,  at 
6  per  cent,  per  annum?  Ans.  $4  27. 

3.  What  is  the  interest  of  1100  dollars,  for  48  days,  at 
6  per  cent,  per  annum?  Ans.  $8  80. 

4.  What  is  the  interest  of  3459  dollars,  for  75  days,  at 
6  per  cent,  per  annum?  Ans.  $43  23  cts.  7  m.-|- 

5.  What  is  the  interest  of  1500  dollars,  for  60  days,  at 
5  per  cent,  per  annum?  Ans.  $12  50. 

CASE  5. 
TJie  amount,  time,  and  rate  per  cent,  given,  to  Jlnd  the 

principal. 

RULE. — Find  the  amount  of  100  dollars  for  the  time 
required,  at  the  given  rate  per  cent. 

Then,  by  proportion,  as  the  amount  of  100  dollars  for 
the  time  required,  (at  the  given  rate  per  cent.)  is  to  the 
amount  given,  so  is  100  dollars  to  the  principal  required. 

EXAMPLES. 

1.  What  principal,  at  interest  for  8  years,  at  5  per  ct. 
per  annum,  will  amount  to  840  dollars?        Ans.  $600. 
5  dollars 
8  years 

40  Int.  of  $100  for  Syr. 
100  $         $        $        $ 

140  :   840  ::100  :  600 
140  Amt.  of  $100  for  8  yr. 


INTEREST.  117 

2.  What  principal,  at  interest,  for  6  years,  at  4  per 
cent,  per  annum,  will  amount  to  $,1240.       Ans.  $1000 

3.  What  principal,  at  interest  for  5  years,  at  6  per  ct. 
per  annum,  will  amount  to  2470  dollars?      Ans.  $1900. 

CASE  6. 

The  principal,  amount,  and  time  given,  to  find  the  rate 
per  cent. 

RULE.  —  Find  the  interest  for  the  whole  time  given,  by 
subtracting  the  principal  from  the  amount. 

Then,  as  the  principal  is  to  100  dollars,  so  is  the  in- 
terest of  the  principal  for  the  given  time,  to  the  interest 
of  100  dollars  for  the  same  time. 

Divide  the  interest  last  found  by  the  time,  and  the 
quotient  will  be  the  rate  per  cent,  per  annum, 

Or  by  compound  proportion. 

EXAMPLES. 

1.  At  what  rate  per  cent,  per  annum,  will  COO  dollars 
amount  to  744  dollars,  in  4  years?  Ans.  6  per  cent. 


$744  amount  As  600  :  100  :  :  144  :  24. 

600  principal          yr.  $ 

4)  24  (6  rate  per  cent. 
144  interest 
Or  by  compound  proportion  : 

$         $ 
As  600  :  100  $          $ 

yr.        yr.     :  :  144    :    6  rate  per  cent. 
4     :       1 

2.  At   what   rate   per  cent,  per  annum,  will  $1200 
amount  to  $1476,  in  5  years  and  9  months? 

Ans.  4  per  cent. 

3.  If  834  dollars,  at  interest  2  years  and  6  months, 
amount  to  927  dollars  82*  cents,  what  was  the  rate  per 
cent,  per  annum?  Ans.  44  per  cent. 


118  COMPOUND  INTEREST. 

CASE  7. 
To  Jlnd  the  time,  when  the  principal,  amount,  and  rate 

per  cent,  are  given. 

RULE. — Divide  the  whole  interest  by  the  interest  of 
the  principal  for  one  year,  and  the  quotient  will  be  the 
time  required,  or  by  proportion. 

EPAMPLES. 

1.  In  what  time  will  400  dollars  amount  to  520  dol- 
lars, at  5  per  cent,  per  annum?  Ans.6  years. 

*  9 

400  520 

5  400 

<&  A  Y      Y 

— — ~  as  jm  i.       JL 


20JOO       20)120(6          20    :    120   : :    1   :  6  Ans. 

2.  In  what  time  will  £1600  amount  to  £2048,  at  4 
per  cent,  per  annum?  Ans.  7  years. 

3.  Suppose  1000  dollars,  at  4t  per  cent,  per  annum, 
amount  to  1281  dollars  25  cents,  how  long  was  it  at  in- 
terest? Ans.6Y.  3rao. 


COMPOUND   INTEREST. 

Compound  interest  is  that  in  which  the  interest  for 
one  year  is  added  to  the  principal,  and  that  amount  is 
the  principal  for  the  second  year;  and  so  on  for  any 
number  of  years. 

RULE. — Find  the  a  mount  of  the  given  sum  for  the  first 
year  by  simple  interest,  which  will  be  the  principal  for  the 
second  year;  then  find  the  amount  of  the  principal  for 
the  second  year  for  the  principal  for  the  third  year;  and 
so  on  for  \  any  number  of  years. 

Subtract  the  first  principal  from  the  amount,  and  the 
remainder  will  be  the  compound  interest  required. 

.    EXAMPLES. 

1.  What  is  the  compound  interest  of  150  dollars  for  5 
years,  at  4  per  cent,  per  annum? 

Ans.  $32,49 


COMPOUND  INTEREST.  119 

$150  $150  t 

4  6          inst  1st  year 

6|00  int.  1  yr.    156          amount  1st  year 
6,24    int.  2d  year 

$156  102,24     amount  2d  year 

4  6,48.9  ink  3d  year 

6|24  168,72,9  amount  3d  year 

6,74.9  ink  4th  year. 

175,47.8  amount  4th  year 
7,01.9  int.  5th  year 

6|48.96  182,49.7  amount  5th  year 

150,00.0  principal 

32,49.7  compound  int.  for  5  years. 

2.  What  is  the  compound  interest  of  760  dollars,  for 
3  years,  at  6  dollars  per  cent,  per  annum? 

Ans.  $145  17  cts.  2  m.-f- 

3.  What  is  the  compound    inierest  of  $242  50   els., 
for  4 years,  at     6  per  cent,  per  annum? 

An.s.  $63  65  cents. 

4.  What  is  the  amount  of  1300  dollars,  for  3  years, 
at  5  dollars  per  cent,  per  annum,  compound  interest? 

Ans.  $1504  91  cts.  2  m.-f 

5.  How   much  is   tha  amount  of  3127  dollars,  for  4 
years,  at  4i  dollars  per  cent,  per  annum,  compound  in- 
terest? Ans.  $3729  OOcts.  5m. 

Questions. 
What  is  interest? 
What  is  the  principal? 
What  is  the  rate  per  cent. 
What  is  the  amount? 

How  do  you  proceed  when  the  interest  for  several 
years  is  required? 


120  COMPOUND    INTEREST. 

What  is  to  be  noted  « f  the  interest  is  required  for 
years  and  months? 

When  the  interest  is  required  for  any  number  of 
weeks  or  days,  less  or  more  than  one  year,  how  do  you 
perform  the  operation? 

How  do  you  proceed  to  find  the  interest,  at  6  per  cent, 
for  any  number  of  daj's,  as  computed  at  banks  ? 

What  is  to  be  observed  when  the  interest  is  at  any 
other  rate  than  6  per  cent.? 

How  do  you  proceed,  when  the  principal,  amount,  and 
time  are  given,  to  find  the  rate  per  cent.? 

How  do  you  find  the  time,  when  the  principal,  amount, 
and  rate  per  cent,  are  given  ? 

What  is  compound  interest  ? 

How  is  compound  interest  computed? 


PROMISCUOUS    EXERCISES. 

1.  What  is  the  interest  of  620  dollars  25  cents  for  5 
years,  at  5i  per  cent,  per  annum? 

Ans.  $170  56  8  m  -f- 

2  What  is   the  interest  of    $420,  for  1  year,  at  7 
per  cent,  per  annum?  Ans.  $29  40. 

3  What  is  the  interest  of  1450  dollars,  for  60  days, 
at  6  per  cent,  per  annum?  Ans.  $14  50  cts. 

4  What  is  the  compound  interest  of  $626  25,  for  3 
years,  at  5i  per  cent,  per  annum? 

Ans.  $103  91.+ 

5  What  is  the  interest  of  $1659  for  3  weeks,  at  4  per 
cent,  per  annum?  Ans.  $3  82i.-f- 

6  In  what  time  will  500  dollars   amount  to  1000  dol- 
lars ut  8  per  cent,  per  annum,  simple  interest? 

Ans.  12  years,  6  months. 

7  What  principal,  at  interest  for  6  years  and  6  months, 
at  2  per  cent,  per  annum,  will  amount  to  250  dollars? 

Ans.  $221  23  cts.  9  m. 

8  At  what  rate  per  cent,  per  annum,  will  $300  amount 
to  $450,  in  5  years?  Ans.  10  per  cent. 


INSURANCE,  COMMISSION,   AND  BROKAGE.  121 

INSURANCE,  COMMISSION,  AND 
BROKAGE. 

INSURANCE,  Commission  and  Brokage,are  allowances 
made  to  insurers,  factors,  and  brokers,  at  such  rate  per 
cent,  as  may  be  agreed  on  between  the  parties. 

RULE. 

Proceed  in  the  same  manner  as  though  you  were  re- 
quired to  find  the  interest  of  the  given  sum  for  one  year. 

EXAMPLES. 

1  What  is  the  commission  on  625  dollars,  at  4  dollars 
per  cent? 

$625 
4 


Ans.  $25,00 

2  What  is  the  commission  on  $1320,  at  5  per  cent.? 

Ans.  $66. 

3  What  is  the  commission  on  3450  dollars,  at  44  dol- 
lars percent.?  Ans.  $155,25. 

4  The  sales  of  certain  goods  amount  to  1680  dollars: 
what  sum  is  to  be  received  for  them,  allowing  2|  dollars 
per  cent,  for  commission?  Ans.  $1633,80. 

5  What  is  the  insurance  of  $760,  at  6i  per  cent.  ? 

Ans.  $49,40. 

6  What  is  the  insurance  of  5630  dollars,  at  7f  dollars 
per  cent.?  Ans.  $436  32  cts.  5  m 

7  A  merchant  sent  a  ship  and  cargo  to  sea,  valued  at 
17654  dollars:  what  would  be  the  amount  of  insurance, 
at  18|  dollars  per  cent.?  Ans.  $3310  12i  cts 

8  What  is  the  brokage  on  2150  dollars  at  2  per  cent.? 

Ans.  $43 

9  When  a  broker   sells  goods  to  the  amount  of  984 
dollars  50  cents,  what  is  his  commission,  at  14  dollar  per 
cent.?  Ans.  $12  30i  cts.-f- 

10  If  a  broker  buys  goods  for  me,  amounting  to  1050 


h 


122  DISCOUNT. 

dollars  75  cents,  what  sum  must  I  pay  him,  allowing  him 
14  per  cent.?  Ans.  $24  76  cts.  1  m.+ 

Questions. 

What  are  Insurance,  Commission,  and  Brokage? 

How  do  you  proceed  to  find  the  Insurance,  Commis- 
sion, or  Brokage? 

In  what  does  this  rule  differ  from  interest?  It  takes 
no  account  of  time. 


DISCOUNT. 

DISCOUNT  is  an  abatement  of  so  much  money  from  any 
sum  to  be  received  before  it  is  due,  as  the  remainder 
would  gain,  put  to  interest  for  the  given  time  and  rate 
per  cent. 

RULE. 

Find  the  interest  of  100  dollars  for  the  given  time  at 
the  given  rate  per  cent. 

Add  the  interest  so  found  to  100  dollars,  then  by  pro- 
portion, 

As  the   amount  of  100  dollars  for  the   given  time, 

Is  to  the  given  sum, 

So  is  100  dollars, 

To  the  present  worth. 

If  the  discount  be  required,  subtract  the  present  worth 
from  the  given  sum,  and  the  remainder  will  be  the  dis- 
count. 

NOTE. — When  discount  is  made  without  regard  to 
time,  it  is  fouuti  precisely  like  the  interest  for  one  year. 

EXAMPLES. 

1  What  is  the  present  worth  of  420  dollars,  due  in  2 
years,  discount  at  6  per  cent,  per  annum? 

Ans.  $375. 


DISCOUNT.  123 

$  $     $     $     r 

6  112  :420::  UK)  .375 

2 

12 
100 

112 

2  What  is  the  present  worth  of  850  dollars,  due  in  3 
months,  at  6  per  cent,  per  annum? 

Ans.  $837  43|  cts.-f 

3  What  is  the  discount  of  645  dollars,  for  9  months, 
at  6  per  cent,  per  annum?  Ans.  $27  77i  cts. 

4  What  is  the  present  worth  of  775  dollars  50  cents, 
due  in  4  years,  at  5  per  cent,  per  annum? 

Ans.  $646,25. 

5  What  is  the  present  worth  of  580  dollars,  due  in  8 
months,  at  6  per  cent,  per  annum?         Ans.  $557,69.-|- 

6  What  is  the  present  worth  of  954  dollars,  due  in  3 
years,  at  4i  per  cent,  per  annum? 

Ans.  $840  52  cts.  8  m.+ 

7  What  is  the  discount  of  205  dollars,  due  in  15 
months,  at  7  per  cent  per  annum? 

Ans.  $16  49  cts.  5  m.+ 

8  Bought  goods  amounting  to  775  dollars,  at  9  months' 
credit:  how  much  ready  money  must  be  paid,  allowing 
a  discount  of  5  per  cent,  per  annum? 

Ans.  $746  98  cts.  7  m. 

9  I  owe  A.  to  the  value  of  1005  dollars,  to  pay  as  fol- 
io^: viz.  475  dollars  in  10  months,  and  the  remainder 
in  To  months;  what  is  the  present  worth,  allowing  dis- 
count at  6  per  cent,  per  annum? 

Ans.  $945  40  cts.  4  m. 

10  What  is  the  difference  between  the  interest  of 
2260  dollars,  at  6  per  cent,  per  annum,  for  5  years,  and 
the  discount  of  the  same  sum  for  the  same  time  and  rate 
percent.?  Ans.  $156  46  cts.  2m.-4- 


124  EQUATION   OF   PAYMENTS. 

11  What   is  the  discount  of  520  dollars,  at   5  per 
cent.? 

$520 
5 


$26,00  Ans.  % 

12  How  much  is  the  discount  of  $782,  at 4  per  cent.? 

Ans.  $31,  28 

13  What  is  the  discount  of  476  dollars,  at  3  per  cent.? 

Ans.  $14,28. 

14  Bought  goods  on  credit,  amounting  to  1385  dol- 
lars :  how  much  ready  money  must  he  paid  for  them,  if 
a  discount  of  6  per  cent,  be  allowed?       Ans.  $1301,90. 

15  I  hold  A.'s  note  for  650  jdollars ;  but  I  agree  to  al- 
low him  a  discount  of  4£  per  cent,  for  present  payment: 
what  sum  mustl  receive?  Ans.  $620,75. 

Questions. 

What  is  discount? 

What  is  first  to  be  done  ? 

After  having  found  the  interest  of  100  dollars,  at  tht 
given  time  and  rate  per  cent.,  what  is  next  to  be  done  ? 

After  having  added  the  interest  so  found  to  100  dol- 
lars or  pounds,  by  what  rule  do  you  work  to  find  the  dis- 
count? 

When  discount  is  made  without  regard  to  time,  how  is 
it  found? 


EQUATION  OF  PAYMENTS. 

EQUATION  is  a  method  of  reducing  several   stated 
times,  at  which  money  is  payable,  to  one  mean,  or  equa- 
ted time,  when  the  whole  sum  shall  be  paid. 
RULE. 

Multiply  each  payment  by  its  time,  and  divide  the  sum 
of  all  the  products  by  the  whole  debt,  the  quotient  will 
be  the  equated  time. 


EQUATION    OF    PAYMENTS.  125 

Proof. — The  interest  of  the  sum  payable  at  the  equa- 
ted time,  at  any  given  rate,  will  equal  the  interest  of  the 
several  payments  for  their  respective  times. 

EXAMPLES. 

1  C.  owes  D.  100  dollars,  of  which  the  sum  of  50 
dollars  is  to  be  paid  at  2  months,  and  50  at  4  months  ; 
but  they  agree  to  reduce  them  to  one  payment;  when 
must  the  whole  be  paid?  Ans.  3  months. 

50X2=100 

50x4=200 


100)300(3  months 

2  A  merchant  hath  owing  to  him  300  dollars,  to  be 
paid  as  follows :  50  dollars  at  2  months,  100  dollars  at  5 
months,  and  the  rest  at  8  months ;  and  it  is  agreed  to 
make  one  payment  of  the  whole;  when  must  that  time 
be?  Ans.  6  months. 

3  F.  owes  H.  2400  dollars  of  which  480  dollars  are 
to  be  paid  present,  900  dollars  at  5  months,  and  the  rest 
at  10  months;  but  they  agree  to  make  one  payment  of 
the  whole,  and  wish  to  know  the  time?     Ans.  6  months. 

4  K.  is  indebted  to  L.  480  dollars  which  is  to  be  dis- 
charged at  4  several  payments,  that  is  i  at  2  months,  i 
at  4  months,  i  at  6  months,  and  i  at  8  months;  but  they 
agreeing  to  make  one  payment  of  the  whole,  the  equa- 
ted time  is  therefore  demanded?  Ans.  5  months. 

5  P.  owes  Q.  420  dollars,  which  will  be  due  6  months 
hence,  but  P.  is  willing  to  pay  him  60  dollars  now,  pro- 
vided he  can  have  the  rest  forborn  a  longer  time :  it  is 
agreed  on;  the  time  of  forbearance  therefore  is  required? 

Ans.  7  months. 

6  A  merchant  bought  goods  to  the  amount  of  2000 
dollars  and  agreed  to  pay  400  dollars  at  the  time  of  pur- 
chase, 800  dollars  at  5  months,  and  the  rest  at  10  months; 
but  it  is  agreed  to  imike  one  payment  of  the  whole;  what 
is  the  mean  or  equated  time?  Ans  6  months. 


11* 


126  BARTER. 

BARTER. 

BARTER  is  the  exchanging  of  one  kind  of  goods  for 
another,  duly  proportioning  their  values,  &c. 

RULE. 

The  questions  that  come  under  this  head,  may  be 
done  by  the  compound  rules,  the  Rule  of  Three,  or 
Practice,  as  may  be  most  convenient. 

EXAMPLES. 

1  A  country  storekeeper  bought  150  bushels  of  salt, 
at  56  cents  per  bushel ;  and  is  to  pay  for  it  in  corn,  at 
33*  cents  per  bushel ;  how  much  corn  will  pay  for  the 
salt?  ct.  ct.  lu.  bu. 

As   33*  :  56  :  :  150  :  252 

OR 
Ct. 

56  334)8400 

150  3  3 


1|00)252K>0 

252  bushels  of  corn. 
Cost  of  the  salt.  8400  cts. 

2  How  much  wheat,  at  1  dollar  25  cents  per  bushel, 
will  pay  for  35  sheep,  at  2  dollars 25  cents  a  piece? 

Ans.63bush. 

3  How  much  sugar,  at  9  cents  per  Ib.  will  pay  for  i 
dozen  pair  of  shoes,  at  1  dollar  75  cents  per  pair? 

Ans.  233*  Ibs. 

4  Row  much  tea,  at  80  cents  per  Ib.  will  pay  for  560 
Ibs.  of  pork,  at  5  cents  per  Ib.?  Ans.  35  Ibs. 

5  Bought  4  hats  for  3  dollars  50  cents;  4  dollars;  4 
dollars  50  cents ;  and  5  dollars — how  much  corn  at  32 
cents   will  pay  for  them?  Ans.  53  bush.  4  qts. 

6  A.  has  420  bushels  of  corn,  which  he  barters  with 
B.  for  cats,  and  is  to  receive  4  bushels  of  oats  for  3  of 
corn — how  many  bushels  of  oats  must  A   receive? 

Ans.  560  bush. 


BARTER.  127 

7  A  boy  bartered  735  pears   for  marbles,  giving  5 
pears   for  2  marbles — how  many  marbles   ought   he  to 
have  received?  Ans.  294  marbles. 

8  A  boy  exchanges  marbles  for  pears,  and  gives  2 
marbles  for  5  pears — how  many  pears  should  he  receive 
for  294  marbles?  Ans.  735  pears. 

9  A  farmer  bartered  3  barrels  of  flour,  at  5  dollars  25 
cents  per  barrel,  for  sugar  and  coffee,  to  receive  an  equal 
quantity  of  each — ho\v  much  of  each  must  he  receive, 
admitting  the  sugar  to  be  valued  at  9  cents  per  Ib.  and 
the  coffee  at  14  cents?  Ans.  68i  Ib.  nearly. 

10  A  bartered  42  hat?,  at  1  dollar  25  cents  per  hat, 
with  B.  for  50  pair  of  shoes,  at  1  dollar  12i  cents   per 
pair — who  must  receive  money,  and  how  much? 

Ans.  B.  $3,75. 

11  Sold  75  barrels  of  herrings,  at  2  dollars  75  cents 
per  barrel,  for  which  I  am  to  receive  75  bushels  of  wheat, 
at  1  dollar  8  cents  per  bushel,  and  the  residue  in  money — 
how  much  money  must  I  receive? 

Ans.  125  dolls.  25  cts. 

12  Sold  35  yards  of  domestic,  at  20  cents*  per  yard, 
and  am  to  receive  the  amount  in  apples,  at  25  cents  per 
bushel — how  many  bushels  must  I  have? 

Ans.  28  bush. 

13  Gave  35  yards  of  domestic  for  28  bushels  of  ap- 
ples, at  25  cents  per  bushel — what  was  the  domestic 
rated  at  per  yard?  Ans.  20  cts. 

14  What  is  rice  per  Ib.  when  340  Ib.  are  given  for  4 
yards  of  cloth,  at  4  dollars  25  cents  per  yard? 

Ans.  5  cts. 

15  Gave  in  barter  65  Ibs.  of  tea  for  156  gallons  of 
rum,  at  33i  per  gallon — what  was  the  tea  rated  at? 

Ans.  80  cts.  per  Ib. 

16  Q.  has   coffee  worth  16  cents  per  pound,  but  in 
barter  raised  it  to  18  cts.;   B.  has  broad  cloth  worth  4 
dollars  64  cents  per  yard — what  must  B.  raise  his  cloth 
to,  so  as  to  make  a  fair  barter  with  Q?        Ans.  $5,23. 


128  LOSS   AND   GAIN. 

17  B.  had  45  hats,  at  4  dollars  per  hat,  for  which  A. 
gives  him  81  dollars  25  cents  in  cash,   and  the  rest  in 
pork,  at  5  cents  per  Ib ;  how  much  pork  will  be  required  ? 

Ans.  1975  Ib. 

18  Two  merchants    barter;  A.  receives  20  cwt.  of 
cheese,  at  2  dollars  87  cents  per  cwt.;  B.  8  pieces  of 
linen,  at  9  dollars  78  cents   per  piece;    which  of  them 
must  receive  money,  and  how  much?     Ans.  A.  $20,84. 

1>9  If  24  yards  of  cloth  be  given  for  5  cwt.  1  qr.  of 
tobacco,  at  5  dollars  7 cents  per  hundred;  what  is  the 
cloth  rated  at  per  yard?  Ans.  $1. 109. 

20  A.  barters  40  yards  of  cloth,  at  98  cents  per  yard, 
with  B.  for  284  Ibs.  of  tea,  at  1  dollar  53  cents  per  Ib. ; 
which  must  pay  balance,  and  how  much  ? 

Ans.  A.  $4,405. 

21  A   has  74  cwt.  of  sugar,  at  8  cents  per  Ib.,  for 
which  B.  gave  him  124  cwt.  of  cheese ,   what  was  the 
cheese  rated  at  per  Ib.'?  Ans.  $.  048. 

22  What  quantity  of  sugar,  at  8  cts.  per  Ib.  must  be 
given  in  barter  for  20  cwt.  of  tobacco,  at  8  dollars  per 
cwt.?  Ans.  17  cwt.  3  qrs.  12  Ib. 

23  P.  has  coffee,  which  he  barters  with  Q.  at  11  cts. 
per  Ib.  more  than  it  cost  him,  against  tea,  which  stands 
Q.  in  1  dollar  33  cents  the  Ib.,  but  he  puts  it  at  1  dollar 
66  cents ;   query,  the  prime  cost  of  the  coffee  ? 

Ans.  $.  443+ 


LOSS  AND  GAIN. 

By  Loss  AND  GAIN,  merchants  and  dealers  compute 
their  gains  or  losses. 

RULE. 

Work  by   the  Compound  Rules,  by  Proportion,   or 
in  Practice,  as  may  be  most  convenient. 


LOSS    AND    GAIN.  '129 

EXAMPLES. 

1  Bought  1234  Ibs.  of  coffee,  at  12j  cts.  per  lb.,  and 
sold  the  whole  for  160  dollars ;   did  I  lose  or  gain  by  it, 
and  how  much?  Ans.  gained  $5,75. 

2  Bought  120  dozen  knives,  at  2  dollars  50  cents  per 
dozen,  and  sold  them  at  18|  cents  a  piece;   did  I  gain 
or  lose,  and  how  much?  Ans.  lost  $30. 

3  Bought  1234  yards  of  muslin,  for  17i  cents,  and 
sold  it  at  20  cents  per  yard;   what  was  the  gain? 

Ans.  $30,85. 

4  Bought  10  chests  of  tea,  each  63  Ibs.  neat,  for  600 
dollars,  and  retailed  it  at  87i  cents  per  lb.;   did  I  gain 
or  lose,  and  how  much  ?  Ans.  lost  48  dol.  75  ct. 

5  Gave  285  dollars  25  cents  for  4564  Ibs.  of  bacon, 
and  sold  it  for  365  dollars  12  cents ;  what  was  the  gain 
per  lb?  Ans.  §.  1|  cts. 

6  Bought  1234  yards  of  muslin,  for  246  dollars  80 
cents,  and  sold  it  for  215  dollars  95  cents;   what  did  I 
lose  per  yard?  Ans.  $.  2i  cts. 

7  Gave  25  cts.  per  bushel  for  corn,  and  sold  it  at  28 
cents;  what  is  the  gain  per  cent.? 

Ans.  12  dolls,  per  100  dolls. 

8  Sold  corn  at  25  cts.  per  bushel,  and  4  cts.  loss; 
what  was  the  loss  per  cent.?  Ans.  $13,79. 

9  Bought  13  cwt.  25  Ibs.  of  sugar,  for  106  dollars, 
and  sold  it  at  9i  cts.  per  lb. ;  what  did  I  gain  per  cent.? 

Ans.  32  dolls.  73  cts. 

10  Bought  128  gallons  of  wine  for  150  dollars,  and 
retailed  it  at  20  cts.  per  pint;   what  was  the  gain  per 
cent.?  Ans.  34  dolls.  40  cts. 

11  Sold  a  quantity  of  goods,  for  748  dollars  66  cents, 
and  gained  10  per  cent;   what  did  I  give  for  them.? 

Ans.  680  dols.  60  cts. 


F  2 


130  LOSS   AND   GAIN. 

dols.     dols.       dols. 
100  110  :  100:   :  748,66 

10  100 


110  110)74866,00($680,60 

12  Sold  goods  to  the  amount  of  $'1234,  and  gained  at 
the  rate  of  20  per  cent.;   what  was  the  prime  cost? 

Ans.  $1028,33* 

13  Soldji  quantity  of  goods,  for  $475,  and  at  a  loss 
of  12  per  cent.;   what  did  1  give  for  them? 

dols.     dols.     dols. 
100  88   :  100  :  :  475 

12  100 

88  88)47500(539,77+Ans. 

14  Sold  hats  to  the  amount  of  $136,  at  20  per  cent, 
loss;  \vhat  was  the  first  cost?  Ans.  $170. 

15  Laid  out  $755  in  salt;   how  much  must  I  sell  it 
for,  so  as  to  gain  12  per  cent.? 

12 
100 


As         100 : :  112  : 755  : :  845,60  Ans. 

16  Bought  32   yards  of  mole  skin  for    128  dollars; 
what  must  I  sell  it  for  per  yard,  so  as  to  gain  20  per 
cent.?  Ans.  4  dols.  80  cts.-j- 

17  Bought  17  yards  of  silk  for  21  dollars;   how  much 
per  yard  /mist  I  Detail  it  for,  and  gain  25  per  cent.  ? 

Ans.  1  dol.  54  cts.-f- 

18  Bought  64  ya.rds  of  muslin  for  1C  dollars  50  cents, 
)>ut  proving  a  bad  bargain,  I  am  willing  to  lose  8  per 
cent;   what  must  I  sell  it  at  per  yard?  Ans.19cts.4m.-f- 

19  When  hats  are  bought  at  48  cents,  and  sold  at  5  i 
cents;   what  is  the  gain  per  cent.?  Ans.  12 i 


LOSS    AND    GAIN.  131 

20  If,  when  cloth  is  sold  for  84  cents  per  yard,  there 
is  gained  10  per  cent.;   what  will  be  the  gain  percent, 
when  it  is  sold  for  1  dollar  2  cents  per  yard? 

Ans.  33  dols.  68  cts.+ 

21  Bought  a  chest  of  tea,  weighing  490  Ibs.  for  $122 
50 ct.  and  sold  it  for  $137  20  cents;  what  was  the  profit 
on  each  lb.?  Ans.  3  cts. 

22  Bought  12  pieces  of  white  cloth,  for  16  dollars  50 
cents  per  piece;  paid  2  dollars  87  cents  a  piece  for 
dying;   for  how  much  must  I  sell  them  each,  to  gain  20 
per  cent.  ?  Ans.  23  dols.  244. 

23  If  28  pieces  of  stuff  be  purchased  at  9  dollars  00 
cents  per  piece,  and  10  of  them  sold  at  14  dollars  40 
cents,  and  8  at  12  dollars  per  piece;   at  what  rate  must 
the  rest  be  disposed  of,  to  gam  10  per  cent. by  the  whole? 

Ans.  5  dols.  568. 

24  Sold  a  yard  of  cloth  for  1  dollar  55   cents,  by 
which  was  gained  at  the  rate  of  15  per  cent.;  but  if  it 
had  been  sold  for  1  dollar  72  cents ;  what  would  have  been 
the  gain  per  cent.?  Ans.  27  dols.  69-j- 

25  If,  when  cloth  is  sold  at  $.  935  a  yard,  the  gain 
is  10  dollars  per  cent. ;  what  is  the  gain  or  loss  per  cent., 
when  it  is  sold  at  80  cents  per  yard  ? 

*Ans.  5  dollars  88+loss. 

26  A  draper  bought  100  yards  of  broad  cloth,  for 
which  he  gave  $56 — I  desire  to  know  how  he  must  sell 
it  per  yard,  to  gain  $19  in  the  whole? 

Ans.  75  ct.  per  yard. 

27  A  draper  bought  100  yards  of  broad  cloth  for  $56; 
I  demand  how  he  must  sell  it  per  yard,  to  gain  $15  in 
laying  out  $100?  Ans.  64  ct.  4  in. 

28  Bought  knives  at  11  cents,  and  sold  them  at  12 
cents;   what  will  I  gain  by  laying  out  100  dollars  in 
knives?       .  Ans.  9  dols.  09+ 

29  Bought  knives  at  11  cents,  and  sold  them  at  12 
cents;   what  did  I  gain  by  selling  to  the  amount  of  100  j 
dollars?  .  Ans.  8  dols.  333+ 


jj  132  FELLOWSHIP. 

11     - 

30  If  by  selling  1  Ib.  of  peppejr  for  10i  cents,  there 
are  2  cents-  lost;  how  much  is  the  loss  per  cent.? 

Ans.  16  dols, 

31  A  merchant  receives   from  Lisbon,   180  casks  of 
raisins,  which  stands  him  in  here  2  dollars  13  cents  each, 
and  by  selling  them  at  3  dollars  68  cents  per  cwt.,  he 
gains  25  per  cent.;  required  the  weight  of  each  cask, 
one  with  another?  Ans.  81  Ib. 


*  FELLOWSHIP. 

FELLOWSHIP  is  a  method  by  which  merchants  and 
others  adjust  the  division  of  property,  loss,  or  gain,  &,c., 
in  proportion  to  their  several  claims. 

CASE  1.     SIMPLE  FELLOWSHIP. 

When  the  claims  are  in  proportion  to  the  amount  of 
stock,  labor,  &c.,  without  regard  to  time. 

RULE.     (By  Proportion.) 
As  the  whole  amount  of  stock  or  labor, 
Is  to  each  man's  portion, 
So  is  the  whole  property,  loss,  or  gain, 
To  each  man's  share  of  it. 

Proof. — The  sum  of  all  the  shares  must  equal  the 
whole  gain,  &,c. 

EXAMPLES. 

1  Two  men  bought  a  stock  of  goods  for  480  dollars, 
of  which  A.  paid  320,  and  B.  160.  They  gained  128 
dollars  by  the  transaction ;  what  was  the  share  of  each  ? 

Ans.  A.  received  85  dols.  33J  cts.  and  B.  42  dollars 
661  cts. 

$       $       $       $  ct.  Proof 

A's.  stock  $320      As  480 : 320 : :  128  :  85,33*       $85,331 
H'p.  stock    160  42,601 

.^ .  &  (ft  (JN  (ft      s%4-  _ 

Whole  st'k  480      As480:160::128  :  42,661         128,00 


FELLOWSHIP.  133 

2  Three  workmen   having  undertaken  to  do  a  piece 
of  work  for  275  dollars,  agreed  to  divide  their  profits  in 
proportion  to  the  amount  of  labor  each  one  performed. 
M.  labored  50  days,  N.  65  days,  and  O.  85  days :  what 
was  the  share  of  each? 

Ans.  M.  received  68  dols.  75  cts. ;  N.  89  dols.  37  i  cts. ; 
aridO.  116  dols.  87  i  cts. 

3  A  merchant  being  deceased,  worth  1800  dollars,  is 
found  to  owe  the  following  sums:  to  A.  1200  dollars,  to 
B.  500  dollars,  to  C.  700  dollars :  how  much  is  each  to 
have  in  proportion  to  the  debt? 

Ans.  A.  900  dols.,  B.  375  dols,  and  C.  525  dols. 

4  Three  drovers  pay  among  them  60  dollars  for  pas- 
ture, into  which  they  put  200  cattle,  of  which  A.  had  50, 
B.  80,  and  C.  70 :  I  would  know  how  much  each  had  to 
pay?  Ans.  A.  15  do!?.,  B.  24  dols.,  C.  21  dols. 

5  A  man  failing,  owes  the  following  sums:  to  A.  120 
dollars,  to  B.  250  dollars  75  cents,  to  C.  800  dollars,  to 
D.208  dollars  25  cents  j  and  his  whole  effects  were  found 
to  amount  to  but  650  dollars :  what  will  each  one  receive 
in  proportion  to  his  demand  ? 

Ans.  A.  $  88.73.+     C.  $221.84.+ 
B.  $185.42.+     D.  $153.99+ 

6  A  bankrupt  is  indebted  to  A.  500  dollars  37  i  cents — 
—to  B.  228  dollars — to  C.  1291  dollars  23  cents — to  D. 
709  dollars  40  cents ;   and  his  estate  is  worth  2046  dol- 
lars 75  cents:   how  much   does  he   pay  per  cent.,  and 
what  does  each  creditor  receive? 

Ans.  He  pays  75  per  cent.,  and  A.  receives  375 
dollars  27!  cts.;  B.  171  dols.,-  C.  G68  dols.  42|  cts.; 
and  D.  532  dols.  5  cts. 

7  If  a  man  is  indebted  to  A.  250  dollars  50  cents,  to 
B.  500  dollars,  to  C.  349  dollars  50  cents,  but  when  he 
comes  to  make  a  settlement,  it  is  found  he  is  worth  but 
960  dollars,  how  much  will  each  one  receive,  if  it  be  in 
proportion  to  their  respective  claims? 

(A.  $218  61  cts.  8  m.+ 

Ans.  {B.  $436  36  cts.  3  m.+ 

(C.  $305  01  ct.    8  m.+ 

12 


134  FELLOWSHIP. 

CASE  2.     COMPOUND  FELLOWSHIP. 

When  the  respective  stocks  are  considered  with  rela- 
tion to  time. 

RULE.     (By  Proportion.) 

Multiply  each  man's  stock  by  its  time;  add  the  several 
products  together;  then: 

As  the  sum  of  the  products 

Is  to  each  particular  product, 

So  is  the  whole  gain  or  loss 
-  To  each  man's  share  of  the  gain  or  loss. 

EXAMPLES. 

1  Three  merchants  traded  together;  A  put  in  120 
dollars  for  9  months,  B.  100  dollars  for  16  months,  and 
C.  100  dollars  for  14  months,  and  they  gained  100  dol- 
lars; what  is  each  man's  share? 

$    mo. 

A's.  stock  120  X  9  =  1080 
B's.  stock  100  X  16  =  1600 
C's.  stock  100  X  14  =  1400 


Sum  4080 

Sum.     Prod.      $          $ 

As  4080  : 1080::  100  :   26,47+    A's.  share. 

As  4080  :  1600  ::100  :   39,214  f  B's.  share. 

As  4080  : 1400  ::100  :  34,31+     C's,  share. 

2  Three  men  traded  together;  L.  put  in  88  dollars  for 
3  months,  M.  120  dollars  for  4  months,  and  N.  300  dol- 
lars for  6  months ;  they  gained  184  dollars:  what  will 
each  man  receive  of  the  gain? 

L.  $  19  09  cts.  4  m 
Ans.     M.  $  34  71  cts.  6  m. 
N.  $130  18  cts.  8  m. 


VULGAR    FRACTIONS.  135 

3  Two  merchants  entered   into  partnership  for   16 
months :    A.  put  in  at  first  $600,  and  at  the  end  of  9 
nonths  put  in  $100  more;  B.  put  in  at  first  $750,  and  at 
lie  end  of  6  months  took  out  $250,  at  the  close  of  the 
time  their  gain  was  $386,  what  was  the  share  of  each? 

Ans.  A's.  share  was  $200,794;  B's.  share  was 
$185,20. 

4  A.,  B.,  and  C.,  made  a  stock  for  12  months;  A.  put 
in  at  first  $873,60,  and  4  months  after  he  put  in  $96,00 
more ;  B.  put  in  at  first  $979,20,  and  at  the  end  of  7 
months  he  took  out  $206,40;  C.  put  in  at  first  $355,20, 
and  3  months  after  he  put  in  $206,40,  and  5  months  after 
that  he  put  in  $240,00  more.     At  the  end  of  12  mouths, 
their  gain  is  found  to  be  $3446,40 ;  what  is  each  man's 
share  of  the  gain? 

(A's.  share  is  $1.334,821 
Ans.  ?B's.      -      -    $1271,61i-f 
(CTs.      -      -      $839,96 

Questions. 

What  is  Fellowship? 

By  what  rule  are  its  operations  performed? 

When  is  Fellowship  simple? 

When  is  it  compound? 

In  what  respect  is  Fellowship  compound? 

Ans.  The  proportion  is  compound :  that  is,  the  divi- 
sion of  property,  gain,  &c.,  is  founded  on  the  compound 
proportion  of  the  stock  and  time. 


VULGAR  FRACTIONS. 

A  VULGAR  FRACTION  is  a  part,  or  parts  of  a  unit  ex* 
pressed  by  two  numbers  placed  one  above  the  other  with 
a  line  between  them.  As  ^,  *,«&c. 

The  number  below  the  line  is  the  denominator,  the 
number  above  the  line  is  the  numerator. 

The  denominator  denotes  the  number  of  parts  into 
which  the  unit  is  divided. 


L36  VULGAR    FRACTIONS. 

The  numerator  shows  how  many  of  those  parts  are  to 

taken. 

Fractions  are  either  proper,  improper,  or  compound. 

A  proper  fraction  is  one  whose  numerator  is  less  than 
its  denominator,  as  f-  or  y. 

An  improper  fraction  is  one  whose  numerator  is 
greater  than  its  denominator,  as  |  or  |. 

A  compound  fraction  is  a  fraction  of  a  fraction,  as  \  of 
3-,  or  |  of  J. 

A  mixed  number  is  a  whole  number  and  a  fraction. 

REDUCTION  OF  VULGAR  FRACTIONS. 

CASE  1. 
To  reduce  a  fraction  to  its  lowest  terms. 

RULE. 

Divide  the  terms  by  any  number  that  will  divide  both 
without  a  remainder,  and  divide  the  quotient  in  the  same 
manner,  and  so  on  till  no  number  greater  than  one  will 
divide  them  :  the  fraction  is  then  at  its  lowest  terms. 

EXAMPLES. 

1.  Reduce  T\4T  to  its  lowest  terms. 
==     result. 


2.  Reduce  —-  to  its  lowest  terms.  Res.  \ 

3.  Reduce  ~~  to  its  lowest  terms.  Res.  — 

4.  Reduce  JJ  to  its  lowest  terms.  Res.  |* 
NOTE.  —  When  a  divisor  cannot  readily  be  found,  divide 

the  denominator  by  the  numerator,  and  that  divisor  by 
the  remainder,  and  so  on,  till  nothing  remain:  the  last 
divisor  is  the  common  measure  of  the  two  numbers  j  with 
which  proceed  as  before. 

5.  Reduce  -f/7  to  its  lowest  terms.  Res.  f- 


VULGAR   FRACTIONS.  137 

5  Reduce  ~~j  to  its  lowest  terms.  Rss.*|. 

85 

85)136(1  Here  17  being  the  last  divisor, 

85  is  the  common  measure  of  85 

-  and  136. 

51)85(1 
51 

34)51(1  j  85  )         (5 

34                            17  -     =  J- 

—  I  136)         (8 
17)34(2 
34 

6.  Reduce  j—  to  its  lowest  terms.  Res.  £. 

7.  Reduce  \\\  to  its  lowest  terms.  Res.  f  . 

8.  Reduce  |f  \\  to  its  lowest  terms.  Res.  i-^. 

~        0 

CASE  2. 

To  reduce  a  mixed  number  to  an  improper  fraction. 
RULE. 

Multiply  the  whole  number  by  the  denominator,  and 
add  the  numerator  to  the  product  for  the  numerator  of 
the  improper  fraction,  and  place  the  denominator  under 
it. 

EXAMPLES. 

1.  Reduce  12  J  to  an  improper  fraction. 

12  Res.  '». 


112  Nine  12's  are  108;  add 

-  4  makes  112  ninths. 

9 

2.  Reduce  17  ^to  an  improper  fraction.    Res.  !f2 

3.  Reduce  45  |  to  an  improper  fraction.    Res.  I|7 

4.  Reduce  24  ~  to  an  imp  roper  fraction.  Res.  4y  T  • 


12* 


138  .         VULGAR  FRACTIONS. 

CASE  3. 
To  reduce  an  improper  fraction  to  its  proper  value. 

RULE. 

Divide  the  numerator  by  the  denominator,  and  the 
quotient  will  be  the  whole  number;  the  remainder,  if 
any,  will  be  the  numerator  of  tho  fraction. 

EXAMPLES. 

1  Reduce  Y  to  its  proper  value.  Res.  3  J. 


5 

2  Reduce  l  J2  to  its  proper  terms.  Res.  12  J. 

3  Reduce  l~2  to  its  proper  terms.  Res.  17  |. 

4  Reduce  4Ty  to  its  proper  terms.  Res.  24  ~. 

CASE  4. 

To  reduce  several  fractions  to  other  fractions  having  a 
common  denominator,  and  retaining  their  value. 

RULE. 

Multiply  each  numerator  into  all  the  denominators 
but  its  own,  for  the  respective  numerators;  and  all  the 
denominators  together,  for  a  common^  denominator. 

EXAMPLES. 
1  Reduce  f-  J  and  f   to  a  common  denominator. 

Res.  4|  f|,  and  ?}. 
2X4X6=48) 
3X3X6=54V  Numerators 
5X3X4=60) 

3X4x6=72,  common  denominator. 
Then  we  have  If  for  f  ;  ||  for  J,  and  4|  for  f  . 
Reduce  each  new  fraction  to  its  lowest  terms,  and  the 
result  will  prove  the  work  to  be  right. 
2.  Reduce  J,  f  ,  and  ^,  to  a  common  denominator 

7>p<3     216      240  J    1  6  8 

1168  •  2JJJ  ana 


3.  Reduce  J,  |,  f  ,  and  T\,  to  a  common  denominator. 

T>Ae     216      288      360      nn,1    252 
,       ^  1X6S-    43  2>    432?    43  2>    attd     4  3;  2' 

4.  Keduce  |,  ^,  |,  and  §,  to  a  common  denominator. 

[_>,lv,      R3  0       480       432      MnJ    50  0 

\  ixes.  -f  .,0,  y-oo,  -7-30-5  ^nci  y^-^. 


VULGAK   TRACTIONS. 


139 


NOTE. — It  is  often  convenient  to  use  the  least  possible 
common  denominator;  to  find  which,  divide  the  denomi- 
nators by  any  number  that  will  divide  two  or  more  of 
them  without  a  remainder,  setting  down  those  that  would 
have  remainders;  then  multiply  all  the  divisors  and  all 
the  quotients  together. 


5      6 


8 


1 


1  51713 

4X3X2X5X7X3=2520  common   denom. 
Which  may  be  divided  separately  by  2,  3,  4,  5,  6, 7,  8, 
and  9,  without  a  remainder. 


1 1 


EXAMPLES. 

5  Find  the  least  common  denominator  for  J,  J, 
T5j,  and  ~,  and  compute  their  equivalent  fractions. 

Res    ---    -i°    2-°    ---    --*. 
240  com.  denom. 
00  X  3=180 
30X  7=210 
20X11=220 
15 X  5=  75 
12  X  9=108 

6  Reduce  J,  |,  |,  T\,  and  ~,  to  their  least  common 
denominator.  Res.  T9~*  V^U  r^l-»  i4^,  and  144. 


2(7 


7  Reduce  f  ,  T\,  T9¥,  |xjto  their  least  common  denom- 

natnr  Poo     15°       168"      »35      onJ    110 

nator.  Kes.  240,  24^,  ^To»  and  ?T¥. 

8  Reduce  the  above  fractions  to  a  common  denomina- 
tor, by  the  general  rule,  Case  4. 


19200        21504 
3-OT20J  - 


14080 
3*120' 


5X10X16X24=19200 

7X  8x16x24=21504 

9X  8X10X24=17280 

11  X  8X10X16=14080 


140  VULGAR  FRACTIONS. 

CASE  5. 
To  reduce  a  compound  fraction  to  a  simple  one. 

RULE. 

Multiply  the  numerators  together  for  a  new  numera- 
tor; and  the  denominators  together  for  a  new  denomina- 
tor. 

EXAMPLES. 
1  Reduce  £  of  £  of  —  to  a  single  fraction. 

Res.  T°¥. 
3X5X9  135  9 


4X6X10  240  16 

2  Reduce  £  of  ~  of  ~  to  a  single  fraction.  Res.  TyT. 

3  Reduce  j  of  £  of  J  to  a  single  fraction.      Res.  -/^. 

4  Reduce  ~J  of  f-  of  i-  to  a  single  fraction.     Res.  TST. 

CASE  6. 

To  reduce  a  fraction  of  one  denomination  to  the  fraction 
of  another  denomination,  but  greater,  retaining  the 
same  value. 

RULE. 

Multiply  the  denominator  of  the  fraction  by  the  num- 
ber of  that  denomination  which  it  takes  to  make  one  of 
the  next,  and  so  on  to  the  denomination  required,  and 
place  the  numerator  of  the  given  fraction  over  it. 

EXAMPLES. 

1  Reduce  f  of  a  quart  to  the  fraction  of  a  bushel.* 

qt. 

2  2        1 

—  =  —        Result,  ¥1T  of  a  bushel. 
3X8X4=9(5      48 

2  Reduce  J  of  an  ounce,  Troy,  to  the  fraction  of  a 
pound.  Res.  T3¥,  or  ~  of  a  pound. 

3  Reduce  j  of  a  nail  to  the  fraction  of  a  yard. 

Res.  ¥\,  or  -2~  of  a  yard. 

3  Reduce  |  of  a  perch  to  the  fraction  of  an  acre. 

Res. 


*  That  is,  what  part  of  a  bushel  are  two-thirds  of  a  quart  ? 


VULGAR    FRACTIONS.  141 

4  Reduce  ~j  of  a  pint  to  the  fraction  of  a  hogshead 

Res.  T^T  of  a  hhd. 

CASE  7. 

To  reduce  ike  fraction  of  one  denomination  to  the  frac- 
tion of  another,  but  less,  retaining  the  same  value. 

RULE. 

Multiply  the  given  numerator  by  the  parts  of  that  be- 
tween it  and  that  to  which  it  is  to  be  reduced,  and  place 
the  product  over  the  given  denominator  for  the  fraction 
required. 

EXAMPLES. 

1  Reduce  j7  of  a  bushel  to  the  fraction  of  a  quart. 

Res.  §  of  a  quart. 

2  Reduce  -^  of  a  yard  to  the  fraction  of  a  nail. 

Res.  J  of  a  nail. 

3  Reduce  y/j  ^  °f  an  acre  to  tne  fraction  of  a  perch. 

Res.  I  of  a  perch. 

4  Reduce  7  —•  of  a  hogshead  to  the  fraction  of  a  pint. 

Res.  ~  of  a  pint. 

5  Reduce  yJg-^  of  a  day  to  the  fraction  of  a  minute. 

Res.  TT  of  a  minute. 

CASE  8. 

To  reduce  a  fraction  to  its  proper  value  or  quantity,  in 
whole  numbers. 

RULE. 

Multiply  the  numerator  by  the  parts  of  the  integer, 
and  divide  by  the  denominator. 

EXAMPLES. 

1  Reduce  J  of  a  yard  to  its  proper  quantity. 

Res.  3  qr.  2  na. 

7  eighths  of  a  yd.  4  eighths  of  a  qr. 

4  4 

8)28  8)16 

3  i  quarters  2  nails 


142  VULGAR   FRACTIONS. 

2  Reduce  £  of  a  pound,  avoirdupois,  to  its  proper 
quantity.  Res.  8  oz.  14§  dr. 

3  Reduce  J  of  a  pound,  Troy,  to  its  proper  quantity. 

Res.  9oz. 

4  Reduce  -•  of  a  mile  to  its  proper  quantity. 

Res.  4  fur.  125  yd.  2  fLlfinch, 

5  Reduce  -f-g  of  an  acre  to  its  proper  quantity. 

Res.  1  rood,  30  pf  ich. 

6  Reduce  J  of  a  dollar  to  its  proper  quantity. 

Res.  GO  cents. 

7  Reduce  ~  of  a  pound  to  its  proper  value. 

Res.  6s.  8d. 

8  Reduce  j~  of  a  year  (365  days)  to  its  proper  quan- 
tity. Res.  225  days. 

9  Reduce  J  of  a  tun  to  its  proper  quantity. 

Res.  3  hhd.  7  gal. 

10  Reduce  J  of  a  ton  to  its  proper  quantity. 

Res.  15  cwt.  2  qr.  6  Ib.  3  oz.  8|  dr 

CASE  9. 

To  reduce  a  given  quantity  to  a  fraction  of  any  greate-i 
denomination  of  the  same  kind, 

RULE. 

Reduce  the  given  quantity  to  the  lowest  denomination 
mentioned  for  a  numerator;  and  th^  integer  to  the  same 
denomination,  for  a  denominator. 

EXAMPLES 

1  Reduce  3  qr.  2  na.  to  the  fraction  of  a  yard. 

Res.  J  of  a  yard, 
qr.     na. 
3        2 
4 

yd.  141         (7 

1X4X4=16)         (8 

2  Reduce  2  roods  20  perches  to  the  fraction  of  an 
acre.  Res.  |  of  an  acre. 


VULGAR  FRACTIONS. 


143 


3  Reduce  6  furlongs  16  poles  to  the  fraction  of  a  mile. 

Res.  j  of  a  mile. 

4  Reduce  9  ounces,  Troy,  to  the  fraction  of  a  pound. 

Res.  f  of  a  pound. 

5  Reduce  7  hours  12  minutes  to  the  fraction  of  a  day. 

Res.  -^g-  of  a  day. 

ADDITION  OF  VULGAR  FRACTIONS. 
RULE. — Reduce  the  given  fractions,  if  necessary,  to 
single  ones,  or  to  a  common  denominator;  add  all  the 
numerators  together,  and  place  the  sum  over  the  com- 
mon denominator. 

EXAMPLES. 

f-  4  W 


- 

7  ft 
3  A 

ry       9 

"    To 


3  I 

7  t- 
8 


19  or  2  f      26  T8o= 


8 


28 


NOTE  1. — When  the  fractions  are  of  different  denom- 
inators, reduce  them  to  a  common  denominator,  and 
proceed  as  above.  (See  Note,  page  139.) 

4  Add  J,  £,  Ti,  T5¥  and- A-  together.  Result  3^\. 
240 


60X  3=180 
30X  7=210 
20X11=22C 
15X  5=  75 
12X  9=108 


IT 


73 

2T7 


^40"  -         2T7 

5  Add  f  ,  f  ,  J,  -&  and  Vi  together. 

6  Add  |,  T\,  TV,  and  II  together, 

7  Add  |,  T3¥,  and  T4¥  together. 

8  Add  f  and  f  together. 

9  Add  -j^,  ii  and  |  together 


Res.  4ii 
Res.  2/-V 

i||=rVo 
Res.  U8T 

Res.  2fVV 


144 


VULGAR    FRACTIONS. 


NOTS  2. — When  mixed  numbers  occur,  place  them  as 
in  examples  2  and  3;  proceed  with  the  fractions  as  di- 
rected in  Note  1 ;  and  if  they  amount  to  one  or  more 
integers,  carry  them  to  the  integers,  and  proceed  as  in 
simple  addition. 

10  Add  5|,  61,  and  41  together.  Res.  l7, 

24 

8x2=16 
3x7=21 
12X1=12 


49) 

— V  =2 

24  \ 


Ti- 


11 Add  21  and  3J  together. 

12  Add  74  and  5£  .together. 


13  Add  171  and  |  together. 

14  Add  4,  6,  9,  £  and  { 


15  Add  5,  7|,  |  and 


T 


together. 
together. 


Result  61. 
Res.  12|J 
Res.  181. 
Res.  2011. 
Res.  13vY 


'    '  2" 

NOTE  3. — When  compound  fractions  are  given,  re- 
duce them  to  single  fractions,  and  proceed  as  before. 
16  Add  TV  of  11,  TCT  of  T\,  and  T7j  of  f  together. 

Res.  j" 
9240  common  denominator 


77X99=7623 
60X^8=-2880 
77X35=2695 


131981 

9240) 

17  Add  |  of  1  and  *  of  1£  together. 

18  Add  1|,  J  of  1,  and  9^  together. 


Res 


Res.  11 


.  - 


19  Add  If0-,  6J,  §  of  1,  and  71  together. 

Res.  16T72\. 

NOTE  4. — When  the  given  fractions  are  of  several 
denominations,  reduce  them  to  their  proper  values  or 
quantities,  and  add  as  in  the  following  example. 


13 


G 


VULGAR    FRACTIONS.  145 

20  Add  £  of  a  pound,  to  ~  of  a  shilling. 

Result  15s.  10rVl. 
15 
s.      d. 


}  of  a  £  15     6$ 
ft  of  a  *.     0     3| 


5x2=10 
3X3=  9 


15  10T4T  19 

— V  =1 
15 

21  Add  2  of  a  pound,  to  J  of  a  shilling. 

Result  18s.  3d. 

22  Add  J  of  a  penny,  to  J-  of  a  pound. 

Res.  2s.  3d.  Iqr.  \, 

23  Add  i  Ib.  troy,  to  ^¥  of  an  ounce, 

Res.  6oz.  lldwt.  16grs. 

24  Add  J  of  a  mile,  to  f$  of  a  furlong.  Res.  Gfu.  28p. 

25  Add  j  of  a  yard,  to  §  of  a  foot.          Res.  2ft.  2  in. 

26  Add  |  of  a  day,  to  |.  of  an  hour.    Res.  8h.  SOmin. 

27  Add  ~  of  a  week,  \  of  a  day,  and  \  of  an  hour 
together.  Res.  2  days,  14  hours,  30min. 

SUBTRACTION  OF  VULGAR  FRACTIONS. 

KT7LE. 

Prepare  the  given  fractions  as  in  Addition;  then  sub- 
tract the  less  from  the  greater,  and  place  the  difference 
over  the  common  denominator. 

EXAMPLES. 

1  Take  |  from  J.  Rem.  ^. 

2  Take  T^  from  f^.  Rem.  {. 

3  Take  T\  from  TV  Rem.  |. 

4  Take  f  from  f .  Rem.  sy 

35 


5x3=15 
7X^=14 


t 

3T 


146  VULGAR   FRACTIONS. 

I         tf         tf         ft 

V    .      Tf    3     V 

T  T2" 2  IT—*  3"  5 

12  com.  cknom  60  com.  denom. 


24  "t  TOO 

From  J  of  a  pound  take  ~  of  a  shilling. 

15  com  denom. 
s.     d.     — 

|  of  a  pound    =15    6|  [    5x2=10 
!\  of  a  shilling  =         3|  |    3x3=9 


s.  15  3Trj        Ans.  jj 
Fron*  |  of  a  £  take  |  of  a  shilling. 

Res.  14s.  3d. 
From  I  of  a  Ib.  troy,  take  1  of  an  ounce. 

Res.  8oz.  IGdwt.  16grs. 

From  1  of  a  yard  take  I  of  an  inch.  Res.  5in.  4. 

From  f  of  a  j£  take  S  of  |  of  a  shilling. 

Res.  10s.  7d.  Iqr.  * 


MULTIPLICATION  OF  VULGAR  FRACTIONS. 

RULE. 

Prepare  the  given  fractions,  if  necessary;  then  mill 
tiply  the  numerators  together  for  a  new  numerator,  and 
the  denominators  together  for  a  new  denominator. 

EXAMPLES. 

1  Multiply  f  by  TV  Res.  T1T 

12    1 

— TT7 — TT 


VULGAR   FRACTIONS.  147 

2  Multiply  TV  by  J.  Res.  ,Y 

3  Multiply  f  by  V-  Res-  2Jf>  or  2T- 

4  Multiply  12J  by  7f .  Res.  96|. 

12|  =  V  and  7f=Y;  then  V  X  V  ='  U9=96l- 

5  Multiply  71  by  8|.  Res.  61|. 

6  Multiply  41  by  1.  Res.   T8T. 

7  Multiply  |- "by  13TV  Res.   12i£. 

8  Multiply  I  of  |  by  T\  of  ft.  Res.  ¥yT. 

9  Multiply  4|  by  f  of  J.  Res.  2f . 

10  Multiply  \  of  7  by  |.  Res.   If. 

11  Multiply  21  by  1*,  and  multiply  the  product  by 
of  |  of  f .  Res.  | 


DIVISION  OF  VULGAR  FRACTIONS. 

RULE. 

Prepare  the  given  fractions,  if  necessary,  then  invert 
the  divisor,  and  proceed  as  in  Multiplication. 

EXAMPLES. 

1  Divide  J  by  1.  Res.  ||. 

8X4=32 

7X9=63 

2  Divide  4  by  f .  Res.  I 

3  Divide  II  by  |.  Res.   Iff 

4  Divide  H  by  4T8¥.  Res.  ^ 

5  Divide  3JJ-  by  9£.  Res.  1 

6  Divide  J  by  4.  Res.  y\ 

7  Divide  4  by  |.  Res.  4| 

8  Divide  1  of  f  by  f  of  J.  Res.  f 

9  Divide  |  of  19  by  f  of  |.  Res.  7| 

10  Divide  4|  by  |  of  4.  Res.  2-^ 

11  Divide  |  of  1  'by  f  of  7|.  Res.  T|T 

12  Divide  5205^  by  J  of  91.  Res.  71| 


148  VULGAR  FRACTIONS. 

PROPORTION  IN  VULGAR  FRACTIONS. 

RULE. 

State  the  question,  (as  in  page  89)  reduce  each  term 
to  its  simplest  form,  invert  the  first  term  or  terms,  and 
proceed  as  in  Multiplication  of  Vulgar  Fractions 

EXERCISES. 

1  If  J  yd.  cost  $f  ;  what  will  |  yd.  cost?     Ans.  50c 
yd.        yd.  D. 

3         .         3         .    .         5       rPI-i/in    4  v  3  y  5  -    "  0    —'I1  -  ^A   /•/« 
i       •       J      •  •       j*     -»  I1(  :I1  3-  A  j  A  -g-  —  Ta'o"  —  *n?  —  *Jv  Ci9i 

2  If  1  bu.  cost  $1  ;  what  will  f  bu.  cost?  Ans.  $2,80. 
bu.     bu.         D. 

1    :    i    ::    1.  Then  <X  JX  J-=y1?=$2f  J=$2,80. 

3  If  A  owned  |  of  a  toll-bridge,  and  sold  f  of  his 
share  for  $681  ;  what  is  the  whole  value?    Ans.  $1520. 

I  of  |  :  V  :  :   6f  4  '  that  is,  -/„   :  1  :  :   "f4.     Then 


4  If  I  barter  5|  cwt.  of  sugar  at  6|  cts.  per  lb.,  for 
indigo  at  $4T5¥  per  lb.  ;  how  much  indigo  must  I  re- 
ceive? Ans.  10  lb.  5  oz.  2j  dr.-f 

D.        cts.         cwt.  cts.         cts.  lb. 

4f-  :  6|  :  :  5»-;  thatis6^0  :  y  :  :  5V6-  Then 
16  4  9'  ¥i|T  X  V  X  5  V3  6  =  YAW  =  10  ^. 
—  —  —  5  oz.  2  dr.  J^J. 

f*         V         ¥ 

5  If  the  cent  roll  weighs  6|  oz.,  when  wheat  is  68| 

cents  per  bu.  ;  what  is  the  cost  of  wheat  per  bu.  when 
it  weighs  4£  oz?  Ans.  $1,03}. 

oz.    oz.  cts. 

41  :  61  :  :  68|;  then  ^X  V  Xajs  =  4}||°  =  103i. 

64  4 

2_5         2_5  2  ^T  5 

6  How  many  men  will  reap  417|  acres  in  121  dayb, 
if  5  men  reap  521  in  61  days?  Ans.  20. 

a.  a.  men. 


1O1      .     «l  5      V    2    V  2  0  8  8  «y  2  5  V  5 

1^1     .     bj  2¥TX25X        J       X     TXy 

NOTE.  —  Tn  multiplying,  omit  the  numbers  that  420  *"i  both  the 
upper  and  lower  series. 


DECIMAL   FRACTIONS.  149 

DECIMAL  FRACTIONS. 

A  decimal  fraction  is  a  fraction  whose  denominator  is 
1,  w^th  as  many  cyphers  annexed  as  there  are  figures  in 
I  the  numerator,  and  is  usually  expressed  by  writing  the 
numerator  only  with  a  point  prefixed  to  it:  thus  T\,  T7^, 
T6o2<ro>  are  decimal  fractions,  and  are  expressed  by  .5, 
.75,  .625. 

A  mixed  number,  consisting  of  a  whole  number  and 
a  decimal,  as  25T\,  is  written  thus,  25.5. 

As  in  numeration  of  whole  numbers  the  values  of  the 
figures  increase  in  a  tenfold  proportion,  from  the  right 
hand  to  the  left;  so  in  decimals,  their  values  decrease  in 
the  same  proportion,  from  the  left  hand  to  the  right, 
which  is  exemplified  in  the  following 

TABLE. 


Whole  numbers.  Decimals. 

NOTE. — Cyphers  annexed  to  Decimals,  neither  in- 
crease nor  decrease  their  value;  thus,  .5,  .50,  .500,  be- 
mg  T5o>  TSO°O>  T\Yo>  are  °f  tne  same  value:  but  cyphers 
prefixed  to  decimals,  decrease  them  in  a  tenfold  propor- 
I  tion;  thus  .5,  .05,  .005;  being  T\,  Tf^,  T¥\^,  are  of  dif- 
ferent values. 


150  DECIMAL   FRACTIONS. 

ADDITION  OF  DECIMALS. 

RULE. 

Place  the  given  numbers  according  to  their  values, 
viz.  units  under  units,  tenths  under  tenths,  &c.,  and  add 
as  in  addition  of  whole  numbers;  observing  to  sot  the 
point  in  the  sum  exactly  under  those  of  the  given  num- 
bers. 

EXAMPLES. 

.12  2.16  .14  .1  .15 

.134  3.45  .24  4.12  .75 

.21  40.02  .122  15.4  .92 

743  35.4  .36  76.36  63.25 

345  36.1  .141  120.16  25. 

.002  125.32  .567  425.04  4. 

1.554        242.45 

6  Add  .5,  .75,  .125,  496,  and  .750  together. 

7  Add  .15,  126.5,  650.17,  940.113,  and  722.2560 
together. 

8  Add  420.,  372.45,   .270,  965.02,  and   1.1756  to- 
gether. 


SUBTRACTION  OF  DECIMALS. 

RULE. 

Place  the  numbers  as  in  addition,  with  the  less  under 
the  greater,  and  subtract  as  in  whole  numbers;  setting 
the  point  in  the  remainder  under  those  in  the  given 
numbers. 

EXAMPLES. 

.4562        56.12          .4314         5672.1  32.456 

.316  1.242         .312  321.12  1.33 

.1402          54.878 

6  From  100.17  take  1.146. 

7  From  146.265  take  45.3278. 

8  From  4560.  take  .720. 


DECIMAL    FRACTIONS.  151 

MULTIPLICATION  OF  DECIMALS. 

RULE. 

Multiply  as  in  whole  numbers,  and  from  the  right 
hand  of  the  product,  separate  as  many  figures  for  deci- 
mals, as  there  are  decimal  figures  in  both  the  factors. 

EXAMPLES. 

1  Multiply  .612  by  4,12      2  Multiply  1.007  by  .041. 
.612  1.007 

4.12  .041 


2.52144 

3  Multiply  37.9  by  46.5 

4  36.5  by  7.27 

5  29.831  by  .952 

6  3.92  by  196. 

7  .285  by  .003 

8  4.001  by  .004 

9  .00071  by  .121 


1007 
4028 

.041287 

Product     1762.35 

265.355 

28.399112 

768.32 

_    .000855 

.016004 

.00008591 


DIVISION  OF  DECIMALS. 


RULE. 

Divide  as  in  whole  numbers,  and  from  the  right  hand 
of  the  quotient,  separate  as  many  figures  for  decimals 
as  the  decimal  figures  of  the  dividend  exceed  those  of 
the  divisor.  If  there  are  not  so  many  figures  as  the 
rule  requires,  supply  the  defect  by  prefixing  cyphers. 


152  DECIMAL   FRACTION*. 

EXAMPLES. 

1  Divide  .863972  by  .92     2  Divide  4.13  by  572.4, 
.92).863972(.9391      572.4)4.130000(.00721+ 
828 .  40068 

359  12320 

276  11448 

837  8720 

828  5724 

92  2996 

92 


3  Divide  19.25  by  38.5  Quotient        .5 

4  234.70525  by  64,25  3.653 

5  1.0012  by  .075  13.34+ 

6  .1606  by  .44  .365 

7  .1606  by  4.4  .0365 

8  .1606  by  44.  .00365 

9  9.  by  .9  10. 

10  .9  by  9.  .1 

11  186.9  by  7.476  25. 

NOTE  1.  When  a  whole  number  is  to  be  divided  by  a 
greater  whole  number,  cyphers  must  be  affixed  to  the 
dividend,  as  decimal  figures. 

12  Divide  3  by  4  Quotient        .75 

13  275  by  3842  .071577+ 

14  210  by*  240  .875 

NOTE  2.  When  any  whole  number  is  divided  by  ano- 
ther, if  there  be  a  remainder,  cyphers  may  be  affixed  to 
the  dividend,  and  the  quotient  continued. 

15  Divide  382  by  25  Quotient         15.28 

16  13689  by  75  182.52 

17  315  by  124  2.5403+ 


DECIMAL    FRACTIONS.  153 

REDUCTION  OF  DECIMALS. 

CASE  1. 

To  reduce  a  vulgar  fraction  to  a  decimal. 
RULE. 

Place  cyphers  to  the  right  of  the  numerator,  until  you 
can  divide  it  by  the  denominator,  and  continue  to  divide 
until  there  is  no  remainder  left;  or  if  it  be  a  number 
which  will  never  come  out  without  a  remainder,  until  it 
is  carried  out  to  a  convenient  number  of  decimal  places. 

EXAMPLES. 

1  Reduce  j  to  a  decimal.         « 

5)40 

— 

.8  Ans. 

2  Reduce  |  to  a  decimal.  Ans.  .875. 

3  Reduce  JJ  to  a  decimal.  Ans.  .70833.-(- 

4  Reduce  T3Ty*  to  a  decimal.  Ans.   .1762.-f- 

5  Reduce  ^f  to  a  decimal.  Ans.  .4566.-J- 

CASE  2. 

To  reduce  any  given  sum  or  quantity  to  the  decimal  of 
any  higher  given  denomination. 

RULE. 

Reduce  the  given  sum  or  quantity  to  the  lowest  de- 
nomination mentioned  in  it. 

Reduce  onet>f  that  denomination  of  which  you  wish 
to  make  it  a  decimal,  to  the  same  denomination  with  the 
given  sum. 

Divide  the  given  quantity  so  reduced  by  one  of  the 
denomination  of  which  you  wish  to  make  it  a  decimal ; 
the  quotient  will  be  the  decimal  required. 


154  DECIMAL   FRACTIONS. 

EXAMPLES. 

1  Reduce  3s.  6d.  to  the  decimal  of  a  pound. 
3a.  6d.=  42        240)42.000(.175  decimals.    Ans, 
£1.      =240  240 


1800 
1680 

1200 
1200 

2  Reduce  2R.  4P.  to  the  decimal  of  an  acre. 

Answer,  .525. 

3  Reduce  2  qr.  2  nails  to  the  decimal  of  a  yard. 

Ans.  .625. 

4  Reduce  5  minutes  to  the  decimal  of  an  hour. 

Ans.    .08333. 

5  Reduce    10  graias   to   the  decimal  of  an   ounce, 
apothecaries'  weight.  Ans.    .02083.-)- 

6  Reduce  2  quarts  1  pint  to  the  decimal  of  a  hogs- 
head. Ans.  .00992.+ 

CASE  3. 
To  reduce  a  decimal  fraction  to  its  proper  value. 

RULE. 

Multiply  the  given  fraction  continually  by  the  denom- 
ination next  lower  than  that  of  which  it  is  a  decimal,  for 
the  proper  value.  • 

EXAMPLES. 

1  What  is  the  value  of  .375  of  a  dollar?  Ans.  37£cts. 

.375 
100 

37.500 
10 

5.000 

2  What  is  the  value  of  .1361  of  a  £.?    Ans.  2s.  8Jdv 


DECIMAL   TRACTIONS.  155 

3  What  is  the  value  of  .235 of  a  day? 

Ans.  5  hours,  38  min.  24  sec. 
I  4  What  is  the  value  of  .42  of  a  gallon? 

Ans.  1  quart,  1.36  pt. 

5  What  is  the  value  of  .253  of  a  shilling?  Ans.  3.036d. 
G  What  is  the  value  of  .436  of  a  yard? 

Ans.  1  qr.  2.976  na. 
7  What  is  the  value  of  .9  of  an  acre? 

Ans.  3R.  24P. 


PROPORTION  IN  DECIMALS. 

RULE. 

State  the  question  as  the  rule  of  three,  in  whole  num- 
bers, only  ohserve,  when  you  multiply  and  divide,  to 
place  the  decimal  points  according  to  the  rules  of  multi- 
plication and  division  of  decimals. 

EXA3IPLES. 

1  If  4.21b  of  coffee  cost  8s.  2.3J.,  what  cost  639.2511).? 
Ib.       Ib.  s.  d.      £   s.  d. 

4.2  :  639.25  :  :  8  2.3  :  62  6  9.49         Ans. 

2  When  1.4  yard  cost  13s.   what,  will  15  yards  come 
to  at  the  same  price?  Ans.  £6  19s.  3d.  1.71  qr. 

3  If  I  sell  1  qr.  of  cloth  for  2  dollars  34.5  cents,  what 
is  it  per  yard?  Ans.  $9  38  cts. 

4  A  merchant  sold  10.5  cwt.  of  sugar,  for  108.30  dol- 
lars, for  which  he  paid  84  dollars  39.12  cents;  what  did 
he  gain  per  cwt.  by  the  sale?       Ans.  $2  27  cts.  7m.-f- 

5  How   many   pieces   of  cloth,  at  .20.8  dollars  per 
piece,  are  equal   in  value  to  240  pieces,  at  12.6  dollars 
per  piece?  Ans.  145.38-J-  pieces. 

6  If,  when  the  price  of  wheat  is  74.6  cents  per  bush- 
el, the  penny  roll  weighs  5.2  oz.,  what  should  it  be  per 
bushel  when  the  penny  roll  weighs  3.5  oz.? 

Ans.  $1  10  cts.  8m.+ 

QueMion. 

How  do  you  perform  operations  in  the  rule  of  three 
in  decimals? 


156  MENSURATION. 

COMPOUND  PROPORTION,  IN  DECIMALS. 

Questions  in  this  rule  are  wrought  as  in  whole  num- 
bers, placing  the  points  agreeably  to  former  directions. 

EXAMPLES. 

1  If  3   men   receive   8.9j£  for  19.5  days  labor,  how 
much  must  20  men  have  for  100.25  days? 

Ans.  305£.  Os.  8.2d. 


i 

If 


19.5:    oO.25  days     '  -  Os' 


2  If  2  persons  receive  1.625s.  for  1  day's  labor,  how 
much  should  4  persons  have  for  10.5  days? 

Ans.  4£.  17s.  lid. 

3  If  the  interest  of  76.5£for  9.5  months,  be  15.24£. 
what  sum  will  gain  6£  in  12.75  months  ? 

Ans.  22£  8s.  9|d. 

4  How  many  men  will  reap  417.6  acres  in  12  days, 
if  5  men  reap  52.2  acres  in  6  days?  Ans.  20  men. 

5  If  a  cellar  22.5  feet  long,  17.3  feet  wide,  and  10.25 
feet  deep,  be  dug  in  2.5  days,  by  6  men,  working  12.3 
hours  a  day,  how  many  days  of  8.2  hours,  should  9  men 
take  to  dig  another,  measuring  45  feet  long,  34.6  wide, 
and  12.3  deep?  Ans.  12  (lays. 


MENSURATION. 

MENSURATION  is  employed  in  measuring  masons'  and 
carpenters'  work,  plastering,  painting  and  paving;  also, 
for  measuring  timber  in  all  its  forms,  and  for  estimating 
quantity  in  length,  superfices,  and  solids,  whenever 
yards,  feet,  inches,  &.C.,  are  employed. 

The  denominations  are,  foot,  inch,  second,  third,  and 
fourth, 

12  Fourths ••,'•  one  1  Third'" 
12  Thirds        one  1  Second" 
12  Seconds     one  1  Inch.  /. 
12  Inches       one  1  Foot.  Ft. 


MENSURATION.  157 

ADDITION. 

RULE. 

Proceed  as  in  Compound  Addition. 

EXAMPLES. 

Ft.  I.     "            Ft.   I.     "  Ft.    I.    "  '"     " 

25     G     3            7'2     4     G  17     9    2  3     11 

14     2    9             54     3     2  18  11   10  8       9 

35  11   10             14     0     8  22  11     5  4       9 

45   10  11             2G     32  14  10  11  10       8 

600             19    0     4  12     0    0  4     10 

490             14     00  10     2840 


132     4     9 

4  Four  floors  in  a  certain  building  contain  each  1084 
feet,  9in.  8"j  how  many  feet  are  there  in  ail? 

Ans.  4339ft.  2  in.  8". 

5  There  are  six  mahogany  boards,  the  first  measures 
27  ft.  3in.,  the  second  25  ft.  llin.,  the  third,  23ft.  lOin., 
the  fourth  20ft.  9in.,  the  fifth  20ft.  Gin.,  and  the  sixth  18 
feet  5  in.;  how  many  feet  do  they  contain? 

Ans.  13tift.  Sin. 

SUBTRACTION. 

RULE. 

Proceed  as  in  Compound  Subtraction. 

EXAMPLES. 

Ft.  I.  "  Ft.  I.  "  Ft.  I.  "  '"  "" 
75  9  9  84  6  4  100  10  8  10  11 
14  6  11  72  9  8  97  2  4  6  8 


61  2  10 


4  If  19ft.  lOin.  be  cut  from  a  board  which  contains 
41ft.  7in.  how  much  will  be  left?  Ans.  21ft.  9in. 

5  Bought  a  raft  of  boards  containing  59621ft.  8in.,  of 
which  are  since  sold  3  parcels,  each  14905ft.  5in.;  how 
many  feet  remain?  Ans.  14905ft.  5in 


14 


158  MENSURATION. 

MULTIPLICATION. 

CASE  1. 
When  the  feet  of  the  multiplier  do  not  exceed  12. 

RULE. 

Set  the  feet  of  the  multiplier  under  the  lowest  denom- 
ination of  the  multiplicand,  as  in  the  following  example  ; 
then  multiply  as  in  Compound  Multiplication,  by  each 
denomination  of  the  multiplier  separately,  observing  to 
place  the  right  hand  figure,  or  number,  of  each  product, 
under  that  denomination  of  the  multiplier  by  which  it  is 
produced. 

EXAMPLES. 
1  Multiply  10  feet  G  inches  by  4  feet  6  inches. 

Product  47  feet,  3  in 
Ft.    I.   " 

10     6  A  table  10  feet  6  inches  long, 

4     6  and  1  foot  wide,  will  make  10  feet 

6  inches,  or  10&  feet,  square  meas- 

530  are. 

42     0  And  4  feet  6  inches,  or  4i  feet 

wide,  will  make  4d  times   10i,  or 

47     3     0  47i  feet,  or  47  feet  3  inches. 


OR  THUS: 
ft.    in. 
10     6 
44  ft. 

5    3 

42     0 

47     3 


NOTE  1. — If  there  are  no  feet  in  the  multiplier,  sup- 
ply their  place  with  a  cypher;  and  in  like  manner  sup- 
ply the  place  of  any  other  denomination  between  the 
highest  and  lowest. 


MENSURATION.                                         159 

10ft.  Gin. 

or  101  feet  long. 

1 

i        i 

1    1    I    1 

1 

1        1 

1    1    1    1 

4fimoa    1  OA   TYinlfA  4^fi 

!     £      | 

1        1 

1    1    1    1 

tlllJtxb      J.V/S      illrtlVt/    TK^li* 

and  itime  104  make  5i  ft. 

1  ' 

1        1 

1    1    1    1 

Added,  make  474. 

CD       | 

1        1 

1    1    1    1 

1 

i        i 

i    i    i    i    I 

Ft. 

7.  "      Ft.  I. 

Ft.    I.    "    •" 

\  2  Multiply  9 

7       by  3    6 

Res.  33    6    6 

3 

3 

11       bv  9     5 

36  10     7 

4 

8 

6  9  by  7     3 

8               62    6    7     9 

5 

28 

10  6  by  3     2 

4               92    2  10     6 

CASE  2. 

When  the  feet  of  the  multiplier  exceed  12. 

RULE. 

Multiply  by  the  feet  of  the  multiplier  as  in  Compound 
Multiplication,  and  take  parts  for  the  inches,  &,c. 

EXAMPLES. 

1  Multiply 

112ft.  3in.  5" 

by  42ft.  4in.  6" 

Ft.    I.     " 

' 

112    3    5 

6X7=42 

G73    8    6 

J 

7 

4 

I 

4715  11     6     '"  "" 

37    5     1 

i     8 

6 

4     8     ] 

L     8    6 

4758     0    9    4    G 

Ft. 

I. 

"         Ft. 

7.     "             Ft.    I.     "  "/ 

2  Multiply  76  7      by     19 

10        Res.  1518  10  10 

3 

127  6      by  184 

8               23545     0  0 

4 

71 

2  6  by     81 

1     8           5777     922 

160  MENSURATION 

APPLICATION. 

1  A  certain  board  is  28ft.  lOin.  6"  long,  and  3ft.  2  in 
4"  wide;  how  many  square  feet  does  it  contain? 

Ans.  92ft.  Sin.  10"  6'-. 

2  If  a  board  be  23ft.  3in.  long,  and  3ft.  Gin.  wide, 
how  many  square  feet  does  it  contain? 

Ans.  81ft.  4in.  6 

3  A  certain  partition  is  82ft.  Gin.  by  13ft.  3in.;  how 
many  square  feet  does  it  contain?    Ans.  1093ft.  lin.  G". 

4  If  a  floor  be  79ft.  Sin.  by  38ft.  llin.,  how  many 
square  feet  are  therein?  Ans.  3100ft.  4in.  4". 

NOTE. — Divide  the  square  feet  by  9,  and  the  quotient 
will  be  square  yards. 

5  If  a  ceiling  be  59ft.  9in.  long,  and  24ft.  Gin.  broad, 
how  many  square  yards  does  it  contain? 

Ans.  lG2yd.  5ft.-4- 
Ft.        I. 

6  in.     *  59        9 

3 

179        3 

8 

1434        0       „ 
29       10       6 

9)14G3       10      6 
162yd.   5ft. 

6  How  many  yards  are  contained  in  a  pavement  50 
feet  9  inches  long,  and  18  feet  4  inches  wide? 

Ans.  115yd.  5ft  5in. 

7  How  many  yards  in  a  ceiling  92ft.  4in.  long,  22ft. 
8in.  wide?  Ans.  232yd.  4ft.  10in.+ 

8  How  many  squares  in  a  floor  37ft.  Gin.  long,  and 
21ft.  9in.  wide?  Ans.  8  squares,  15  feet.-f- 

A  square  is  10  feet  long  and  10  feet  wide,  or  100 
square  feet.  It  is  used  in  estimating  flooring,  roofing, 
weather-boarding,  &-c. 

9  How   many   squares  of  weather-boarding  on  the 
side  of  a  house  43  feet  6  in.  long,  and  18ft.  Sin.  high? 

Ans.  8  squares,  12ft. 


MENSURATION. 


161 


10  How  many  squares  in  a  roof  36ft.  4in.  long,  15ft. 
9in.  wide?  Ans.  5  sq.  72ft.-f 

NOTE  2. — To  measure  a  triangle.  Multiply  the  base 
by  one  half  the  perpendicular  height,  and  the  product 
will  be  its  superficial  content. 

11  Let  C,  H,  and  G,   represent  a  triangle,  whose 
base  is  40  feet,  and  perpendicular  height  28  feet;  how 
many  feet  does  it  contain  ?  Ans.  560  feet. 


40  feet 


40  feet 


feet. 
40 
14  half  the  perpendicular 

160 
40 

560 

12  How  manv  square  feet  in  a  triangle  80  feet  long 
and  36  feet  high?  Ans.  1440ft. 

13  In  a  triangular  pavement  46  feet  long,  and  24  feet 
at  the  place  of  its  greatest  width,  how  many  yards ;  and 
how  many  bricks,  allowing  41  to  every  square  yard? 

Ans.  61yd.  3ft.,  and  2514  bricks 

14  In  the  gable  ends  of  a  house,  which  is  63  feet  long 
and  22  feet  high,  from  the  "square  of  the  building"  to 
the  top,  how  many  squares?  Ans.  6sq.  93in 

NOTE  3, — To  find  the  circumference  of  a  circle,  when 
the  diameter  is  given:  Say, 

As  7  are  to  22,  so  is  the  diameter  to  the  circumfer- 
ence ;  or  the  contrary, 

As  22  are  to  7,  so  is  the  circumference  to  the  diam- 
eter. 


U* 


162  MENSURATION. 

The  diameter  of  a  circle   is  14 
feet;  what  is  the  circumference? 

Ans.  44. 

As  7  :  22  :  :  14   :  44 
The   circumference  of  a  circle 
is  44ft.;  what  is  the  diameter? 

Ans.  14. 


NOTE  4. — To  find  the  superficial  contents  of  a  circle. 
Multiply  half  the  circumference  by  half  the  diameter 

15  How  many  square  feet  in  a  circle  whose  diameter 
is  14  feet,  and  circumference  44?  Ans  154  ft. 

half  circumference,  22 

half  diameter,  x       7 

154  feet 

16  How  many  square  feet  in  a  circle  whose  circum- 
ference is  16  feet?  Ans.  20  sq.  ft. 

halfcir.     8 

As  22  :  7::  16  :  5  halfdiam.2i 

—20 

17  How  many  square  feet  in  a  circle  whose  diameter 
is  21  feet?  Ans.  346*  ft. 

NOTE  5. — To  find  the  superficial  contents  of  a  globe. 
Multiply  the  circumference  by  the  diameter. 

18  What  are  the  superficial  contents  of  a  globe  whose 
diameter  is  70  feet,  and  circumference  220  feet? 

Ans.  15400  sq.  ft. 

19  How  many  square  feet     of  cloth  would  be  re- 
quired to  cover  a  globe,  whose  diameter  is  28  feet,  and 
circumference  88?  Ans.  2464  ft. 

20  How  many  yards  of  canvass  would  be  required  to 
make  a  balloon  of  a  globular  form,  20yardsin  diameter? 

Ans.  1257  sq.  yds. 

NOTE  6. — To  find  the  solid  contents  of  a  cube,*  or  of 
a  square  stick  of  timber,  or  a  .pile  of  wood,  fyc. 
Multiply  the  length  by  the  breadth,  and  that  product  by 
the  thickness. 


*  A  cube  is  a  solid  body,  contained  by  6  equal  sides,  ail  of  which  are 
exact  squares. 


3UENSUKATION 


163 


21  What  are  the  solid  contents  of  a  cube  whose  di- 
ameter is  4  feet?  Ans.  64  feet. 

4  feet 
4 

16 

4 

64 

22  What  is  the  solid  contents  of  a  stick  of  timber  2 
feet  thick,  3  feet  wide,  and  36  feet  long? 

Ans.  216  solid  feet. 
36  feet 
3 

108 
2 

216 

23  How  many  solid  feet  in  a  block  of  marble  3  feet 
thick,  7  feet  wide,  and  13  feet  long?       Ans.  273  sol.  ft. 

24  In  a  cube  whose  diameter  is  7  feet,  how  many  solid 
feet?  Ans.  343  feet. 

25  How   many  solid  feet  in  a  pile  of  wood  28  feet 
long,  8  feet  wide,  and  10  feet  high;  and  how   many 
cords  does  it  contain*?       Ans.  2240  feet;  17  cords  64  ft. 

or,  17  i  cords. 

26  In  a  cellar  36  feet  long,  27  feet  wide,  and  4i  feet 
deep,  how 'many  solid  yards?  Ans.  162  yards. 

27  How  many  perches*  of  stone  in  a  wall  42  feet 
long,  84  feet  high,  and  2  feet  thick?         Ans.  28,8  per. 

feet 
24,75)714.00(28,8 

28  In  a  12  inch  brick  wall,  52  feet  long  and  36  feet 
high,  how   many  bricks,  allowing  21  to  every  square 
foot  of  wall?  Ans. 


*  A  perch  is  16*  feet  long,  1£  ft.  wide,  and  1  foot  hig! 
solid  feet. 


164  MENSURATION. 

29  In  an  8  inch  brick  wall,  82  feet  long  and  16  feet 
high;  how  many   bricks,  allowing  14  bricks  for  every 
square  foot  of  wall?  Ans.  18368. 

30  In  a  16  inch  brick  wall,  148  feet  long  and  42  feet 
high,  how  many  bricks,  allowing  28  bricks  to  the  square 
foot?  Ans.  174048. 

31  How  many  bricks  in  3  walls,  the  first  68  feet  long. 
18  feet  6  inches  high,  16  inches  thick;  the  second  72  ft. 
6in.  long,  19ft.  4in.  high,  12in.  thick;  the  third  43ft.  4in. 
long,  12ft.  Sin.  high,  8in.  thick?  Ans.  72343.+ 

NOTE  7. — To  find  the  solid  contents  of  a  cylinder.* — 
Find  the  contents  of  one  end  by  Note  4,  and  multiply 
that  product  by  the  length. 

32  What  are  the  solid  contents  of  a  cylinder  whose 
diameter  is  14  feet,  arid  length  16  feet?       Ans.  2464ft. 

half  circumference      22 
half  diameter  7 


2464  feet 

33  What  are  the  contents  of  u  circular  well,  7  feet  in 
diameter,  and  62  feet  deep?  Ans.  2387  ft. 

34  What  are  the  solid  contents  of  a  tub  whose  diam- 
eter is  6  feet  and  height  7  feet?  Ans.  198  ft. 

35  How  many  feet  in  a  circular  well,  10  feet  diame- 
ter and  20  feet  deep  ?  Ans.  157y  feet. 

NOTE  8. — To  find  the  solid  contents  of  the  f rust  rum 
of  a  cone,  To  the  sum  of  the  squares  of  the  two  diame- 
ters, in  inches,  add  their  product ;  multiply  this  sum  by 
one-third  of  the  depth,  and  this  last  product  by  the  deci- 
mal .7£54.  Eu.  El. 

The  result  will  be  the  contents  in  cubic  inches,  which 

•***'•     »s  desired. 


*  A  c  /Under  is  a  long  round  body,  whose  diameter  is  every 
where  tk.  e  same,. 


MENSURATION.  165 

The  liquid  gallon  of  Ohio  contains  231  cubic  inches. 
The  dry  gallon  contains  268i  cubic  inches. 
The  bushel,  of  grain,  contains  2150|  cubic  inches. 
The  bushel,  of  coal,  lime,  &c.  2688  cubic  inches. 

EXAMPLES. 

1  In  a  circular  vessel,  whose  greater  diameter  is  80 
inches,  the  less  71,  and  the  depth  34,  what  is  the  contents 
in  liquid  gallons  ;  and  also  in  bushels  of  grain  ? 

A        C  659.72-1- gallons. 
3*  I    70.86  bushels. 

80X80=6400=  square  of  80. 
71X71  =  5041=  square  of  71. 

11441 
80X71=5680=  product  of  80  and  71. 

17121 

£  in  depth     =     lU=j  of  the  depth. 


194038X. 7854=152397.4452  inches. 
Divide  by  231  for  gallons,  and  2150|  for  bushels. 

2  The  greater  diameter  of  a  tub  is  38  inches,  the  less 
20.2,  and  the  depth  21 ;  what  is  the  content  in  gallons  ? 

Ans.  62. 34+ gallons. 

3  The  top  diameter  of  a  tube  is  22  inches,  the  bot- 
tom 40,  and  the  height  60 ;  what  is  its  contents  in  gal- 
lons, also  in  bushels  of  grain  ?       .        C  20 1.55-1- gals. 

*'l    21.64+bu. 

4  How  many  barrels,  of  32  gallons  each,  in  a  cistern, 
whose  greater  diameter  is  8  feet  6  inches,  the  less  8  feet, 
and  depth  7  feet  9  inches  ?     Ans.  96  barrels  28-j-gals. 

5  How  many  bushels  of  grain  in  i  bin  that  is  8  feet 
long,  4  feet  wide,  and  6  feet  high  ?         Ans.  123-{-bu. 

6  How  many  bushels  of  coal  in  a"  boat  60  feet  long, 
16  feet  wide,  and  4%  feet  deep  ?     Ans  2777+bushels. 


1 66  MENSURATION. 

NOTE  9. — To  find  the  solid  contents  of  a  round  stick 
of  timber  of  a  taper  from  one  end  to  the  other.  Find  I 
the  circumference  a  little  nearer  the  larger  than  the 
smaller  end;  from  this,  by  Note  3,  find  the  diameter: 
multiply  half  the  diameter  by  half  the  circumference, 
and  the  product  by  the  length.* 

EXAMPLES. 

1  What  are  the  solid  contents  of  a  round  stick  of  tim- 
ber 10  feet  long,  and  2.61  feet  circumference? 

Ans,  5.4  feet.+ 
As  22  :  7  :  :  2.61  :  .83  diameter 

1,305  half  circumference 
.415  half  diameter 

6525 
1305 
5220 

.541575 

10  length 


5.415750 

2  How  many  solid  feet  in  a  log  40  feet  long,  which  girts 
66  inches?  Ans.  96.25 ft. 

As  22  :  7  : :  66  :  21  in  diameter 

33  xlOiX40=  13860.  144)13860(96.25 

NOTE  10. — To  find  the  solid  contents  of  a  globe. — 
Multiply  the  cube  of  the  diameter  by  .5236. 

EXAMPLES. 

1  What  are  the  solid  contents  of  a  globe  whose  diame- 
ter is  14  inches?  Ans.  1436.75in.+ 

14  X 14  X  14=2744.       2744  X  .5236=  1436.7584. 

2  What  are    the  contents  of  a  balloon  of  a  globular 
form,  42  feet  in  diameter?  Ans.  38792.4  ft.-{- 

3  How  many  solid  miles  are  contained  in  the  earth,  or 
globe,  which  we  inhabit? 

*  This  method,  thoitgli  not  quite  accurate,  is  sufficiently  near  the 
truth  for  the  purpose  of  measuring  timber. 


INVOLUTION.  167 

Suppose  the  diameter  to  be  7954  mhes:  then,  7954  X 
7954x7954=503218686664  the  cube  of  the  earth's 
axis,  or  diameter;  then, 

50321 8686664  X  .5236=2634853(34337 

cubick  miles.     Ans. 

NOTE. — The  solidity  of  a  globe  may  be  found  by  the 
circumference,  thus — Multiply  the  cube  of  the  circum- 
ference by  .016887— the  product  will  be  the  contents. 


INVOLUTION,  OR  THE  RAISING  OF 
POWERS. 

The  product  arising  from  any  number  multiplied  by 
itself,  any  number  of  times,  is  called  its  power,  as  fol- 
lows t 

2x2=  4  the  square,  or  2d  power  of  2. 
2x2x2=  8  3d  power  or  cube  of  2. 
2X2X2X2=16  4th  power  of  2. 
The  number  which  denotes  a  power  is  called  its  index. 

NOTE. — When  any  power  of  a  vulgar  fraction  is  re- 
quired, first  raise  the  numerator  to  the  required  "power, 
and  then  the  denominator  to  the  required  power,  and 
place  the  numerator  over  the  denominator  as  before : 

thus,  the  4th  power  of  | 


Questions. 

What  is  the  product,  arising  from  the  multiplication  of 
any  figure  by  itself  a  given  number  of  times,  called? 

What  is  the  number  which  denotes  a  power,  called  ? 

How  do  you  proceed  to  find  any  required  power  of  a 
vulgar  fraction? 


168 

INVOLUTION. 

i 

Table  of 

the  first  nine  Powers.                  \ 

„ 

8s 

£ 

] 

| 

6 

II 

1 

1 

e, 

i 
i 

1 

1 

1 

1      1 

1 

1 

1 

1 

1 

i 

2    4 

8 

16 

32 

64 

128 

256 

512 

3   9 

416 
525 
63fi 

749 
864 
981 

27 
64 
125 
216 
343 
512 
729 

81 
256 
625 
1296 
2401 
4096 
6561 

243 
1024 
3125 
77-76 
16807 
32768 
59049 

729 
4096 
15625 
46656 
117649 
•262144 
531441 

2187 
16384 
78125 
279936 
823543 
2097152 
4782969 

6561 
65536 
390625 
1679616 

5764801 
16777216 
43046721 

19683   - 
262144 
1953125 
10077696 
40353607 
134217728 
387420489 

EXAMPLES. 

1  What  is  the  square  of  32? 
32 

64 
96 

1024  Ans 
2  What  is  the  cube  of  14? 
14 
14 

Ans.  2744. 

56 

196 
14 

784 
196 

2744 
3  What  is  the  sixth  power  of  2.8?    Ans   481.890304 
4  What  is  the  third  power  of  .263?  Ans.  .018191447. 

TUB  SQUARE  ROOT.  169 

EVOLUTION,  OR  THE  EXTRACTING  OF 
ROOTS. 

The  root  of  a  number,  or  power,  is  such  a  number,  as 
being  multiplied  into  itself  a  certain  number  of  times, 
will  produce  that  power,  Thus  2  is  the  square  root  of 
4,  because  2x2=4;  and  4  is  the  cube  root  of  64,  be- 
cause 4x4X4=64,  and  soon. 


THE  SQUARE  ROOT. 

The  square  of  a  number  is  the  product  arising  from 
that  number  multiplied  into  itself. 

Extraction  of  the  square  root  is  the  finding  of  such  a 
number,  as  being  multiplied  by  itself,  will  produce  the 
number  proposed.  Or,  it  is  finding  the  length  of  one 
side  of  a  square. 

RULE. 

1  Separate  the  given  number  into  periods  of  two  fig- 
ures, each,  beginning  at  the  units  place. 

2  Find  the  greatest  square  contained  in  the  left  hand 
period,  and  set  its  root  on  the  right  of  the  given  number: 
subtract  said  square  from  the  left  hand  period,  and  to  the 
remainder  bring  down  the  next  period  for  a  dividual. 

3  Double  the  root  for  a  divisor,  and  try  how  often  this 
divisor  (with  the  figure  used  in  the  trial  thereto  annexed) 
is  contained  in  the  dividual:  set  the  number  of  times  in 
the  root;  then,  multiply  and  subtract  as  in  division,  and 
bring  down  the  next  period  to  the  remainder  for  a  new 
dividual. 

4  Double  the  ascertained  root  for  a  new  divisor,  and 
proceed  as  before,  till  all  the  periods  are  brought  down. 

NOTE. — If,  when  all  the  periods  are  brought  down,  there  be  a  re- 
mainder, annex  cyphers  to  the  given  number,  for  decimals,  and  pro- 
ceed till  the  root  is  obtained  with  a  sufficient  degree  of  exactness. 

Observe  that  the  decimal  periods  are  to  be  pointed  off  from  the  de- 
cimal point  toward  the  right  hand :  and  that  there  must  be  as  many 
whole  number  figures  in  the  root,  as  there  are  periods  of  whole  num- 
bers, and  as  many  decimal  figures  as  there  are  periods  of  decimals. 


15 


170  THE   SQUARE   ROOT. 

PROOF. 

Square  the  root,  adding  in  the  remainder,  (if  any,) 
and  the  result  will  equal  the  given  number. 
EXAMPLES. 

1  What  is  the  square  root  of  5499025? 

5,49,90,25(2345  Ans. 

4  2345 

2345 

43)149  

129  11725 

9380 

464)2090  7035 

1856  4690 

4685)23425  5499025  Proof. 

23425 

2  What  is  the  square  root  of  106929?  Ans.  327. 

3  What  is  the  square  root  of  451584?  Ans.  672. 

4  What  is  the  square  root  of  36372961?  Ans.  6031. 

5  What  is  the  square  root  of  7596796? 

Ans.  2756.£2S+ 

6  What  is  the  square  root  of  3271.4007? 

Ans.  57.19+ 

7  What  is  the  square  root  of  4.372594? 

Ans.  2.091+ 

8  What  is  the  square  root  of  10.4976?         Ans.  3.24 

9  What  is  the  square  root  of  .00032754? 

Ans.  .01809+ 

10  What  is  the  square  root  of  10?        Ans.  3.16224 

To  extract  the  Square  Root  of  a  Vulgar  Fraction. 

RULE. 

Reduce  the  fraction  to  its  lowest  terms,  then  extract 
the  square  root  of  the  numerator  for  a  new  numerator, 
and  the  square  root  of  the  denominator  for  a  new  deno- 
minator. 

NOTE. — If  the  fraction  be  a  surd,  that  k,  one  whose 
root  can  never  be  exactly  found,  reduce  it  tD  a  decimal, 
and  extract  the  root  therefrom. 


THE   SQUARE   ROOT.  171 

EXAMPLES. 

1  What  is  the  square  root  of  |54f  •  J^ns- 1- 

2  What  is  the  square  root  of  f  14-j-?  Ans.  J. 

3  What  is  the  square  root  of  flf  ?         Ans.  .93309-[- 

To  extract  the  Square  Root  of  a  Mixed  Number. 

RULE. 

Reduce  the  mixed  number  to  an  improper  fraction, 
and  procee'd  as  in  the  foregoing  examples :  or, 

Reduce  the  fractional  part  to  a  decimal,  annex  it  to 
the  whole  number,  and  extract  the  square  root  there- 
from. 

EXAMPLES. 

1  What  is  the  square  root  of 37J|?  Ans.  61. 

2  What  is  the  square  root  of  27^?  Ans.  51. 

3  What  is  the  square  root  of  Soli?          Ans.  9.27-|- 

APPLICATION. 

1  The  square  of  a  certain  number  is  105G25:  what 
is  that  number?  Ans.  325. 

2  A  certain  square  pavement  contains  20736  square 
stones,  all  of  the  same  size;  what  number  is  contained 
in  one  of  its  sides?  Ans.  144. 

3  If  484  trees  be  planted  at  an  equal  distance  from 
each  ether,  so  as  to  form  a  square  orchard,  how  many 
will  be  in  a  row  each  way?  Ans.  22. 

4  A  certain  number  of  men  gave  30s«  Id.  for  a  chari- 
table purpose;  each  man  gave  as  many  pence  as  there 
were  men:  how  many  men  were  there)  Ans.  19. 

5  The  wall  of  a  certain  fortress   is  17  feet  high, 
which  is  surrounded  by  a  ditch  20  feet  in  breadth;  how 
long  must  a  ladder  be  to  reach  from  the  outside  of  the 
ditch  to  the  top  of  the  wall?  Ans.  26.24-J-feet. 

NOTE. — The  square  of  the 
longest  side  of  a  right  angled 
triangle  is  equal  to  the  sum  of 
the  squares  of  the  othef  two 
sides ;  and  consequently,  the 
difference  of  the  square  of  the 


longest,  and  either  of  the  other,  Ditch, 

is  the  square  of  the  remaining  one. 


172  THE   SQUARE    ROOT. 

6  A  certain  castle  which  is  45  yards  high,  is  surroun- 
ded by  a  ditch  60  yards  broad ;  what  length  must  a  ladder 
be  to  reach  from  the  outside  of  the  Hitch  to  the  top  of  the 
castle  ?  Ans.  75  yards. 

7  A  line  27  yards  long,  will  exactly  reach  from  the 
top  of  a  fort  to  the  opposite  bank  of  a  river,  which  is 
known  to  be  23  yards  broad ;  what  is  the  height  of  the 
fort?  Ans.  14.1.42-|-yards. 

8  Suppose  a  ladder  40  feet  long  be  so  planted  as  to 
reach  a  window  33  feet  from  the  ground,  on  one  side  of 
the  street,  and  without  moving  it  at  the  foot,  will  reach  a 
window  on  the  other  side  21  feet  high;  what  is  the  breadth 
of  the  street?  Ans.  56.64+feet. 

9  Two  ships  depart  from  the  same  port;  one  of  them 
sails  due  west 50  leagues,  the  other  due  south  84  leagues; 
how  far  are  thev  asunder?? 

Ans.  97.75+  Or,  97|+leagues. 


Questions. 

What  is  a  square?  A  square-  is  a  surface  whose 
length  and  breadth  are  equal,  and  whose  angles'  (or  cor- 
ners) ore  right  angles,  (or  square.) 

What  is  its  square  root?  The  square  root  is  the 
length  of  the  side  of  a  square. 

If  the  square  be  sixteen,  what  is  the  root? 

Why  is  the  root  four? 

If  the  root  be  three,  what  is  the  SQUARE? 

What  is  the  square  root  of  twenty-five? 

What  is  the  square  of  five  ? 

What  is  the  square  root  of  thirty-six  ? 

What  is  the  square  of  six? 

How  do  you  point  off  a  number  whose  square  root  is 
to  be  extracted? 

What  is  the  next  step?  What  do  you  subtract  from 
the  period?  What  do  you  annex*  to  the  remainder? 


THE    SQUARE   ROOT. 


173 


Illustration  of  the  Rule  for  extracting  the  Square  Root. 

The  reason  for  pointing  off  the  given  number  into 
periods  of  two  figures  each,  is,  that  the  product  of  any 
whole  number  contains  just  as  many  figures  as  are  in 
both  the  multiplier  and  the  multiplicand,  or  but  one  less ; 
consequently,  the  square  contains  just  double  as  many 
figures  as  the  root,  or  one  less. 

A  E         B 

Suppose  the  figure  ABCD 
contains  1849  square  feet, 
and  that  the  number  consists 
of  two  periods;  then  there 
must  be  two  figures  in  the 


D 


120 


11 


1600 


9 


120 


root. 

The  largest  root  whose 
square  can  be  taken  out  of 
the  left  hand  period,  is  4,  (or 
as  it  will  stand  in  ten's  place 
in  the  root,  it  is  40,)  and  the 
square  of  this  is  16  (of  1600.) 
This  taken  from  the  whole 
C  square  ABCD,  or  1849, 
leaves  249. 


18,49(43 
16 

83)249 
249 


GIRD 
AEIIG 
HFCI 
EBFH 

ABCD 


Now  double  GH  or  HI, 
which  is  40,  for  a  divisor, 
omitting  the  cypher  to  leave 
place  for  the  next  quotient 
figure,  to  complete  the  divi- 
sor. 

80  into  249  are  contained 
3  times ;  this  3  is  the  width 
of  the  oblong  ALHG,  or 
HFCI.  But  the  square  is 
imperfect  without  EBFH; 
then  annex  the  three  to  the 
divisor.  Now  multiply  this 
perfect  divisor  by  the  last 
figure  of  the  root,  to  get  the 
=  1849  Quantity  m  tne  two  oblong 
figures,  and  the  small  square 
which  comprises  the  great 
square  ABCD. 


15* 


174  THE    CUBE    ROOT. 

How  do  you  find  the  divisor? 
Why  do  you  place  the  new  quotient  figure  in  the  units 
place  of  the  divisor? 

How  do  you  prove  the  square  root? 


THE  CUBE  ROOT. 

The  cube  of  a  number  is  the  product  of  that  number 
multiplied  into  its  square  ? 

Extraction  of  the  cube  root  is  finding  such  a  number 
as,  being  multiplied  into  its  square,  will  produce  the 
number  whose  cube  root  is  extracted. 


RULE. 

Separate  the  given  number  into  periods  of  three  fig 
ures  each,  beginning  at  the  units  place.  Find  the  great- 
est cube  in  the  left  hand  period,  and  set  its  root  in  the 
quotient;  subtract  said  cube  from  the  period,  and  to  the 
remainder  bring  down  the  next  period  for  a  dividual. 

Square  the  root,  and  multiply  the  square  by  three 
hundred  for  a  divisor. 

See  how  often  the  divisor  is  contained  in  the  dividual, 
and  place  the  result  in  the  quotient. 

Multiply  the  divisor  by  the  last  found  quotient  figure ; 
square  the  last  found  figure — multiply  the  square  by  the 
preceding  figure  or  figures  of  the  quotient,  and  this  pro- 
duct by  thirty;  and  cube  the  last  figure.  Add  these 
three  products  together,  and  subtract  their  amount  from 
the  dividual. 

To  the  remainder  add  the  next  period,  and  proceed  as 
before,  until  the  periods  are  all  brought  down. 

When  a  remainder  occurs,  annex  periods  of  cyphers 
to  obtain  decimals,  which  may  be  carried  to  any  conve- 
nient number. 

NOTE  1. — The  cube  root  of  a  vulgar  fraction  is  found 
by  reducing  it  to  its  lowest  terms,  and  extracting  the  root 
rf  the  numerator  for  a  numerator,  and  of  the  denomica- 


THE    CUBE    ROOT.  175 

tor  for  a  denominator.     If  it  be  a  surd,*  extract  the  root 
of  its  equivalent  decimal. 

EXAMPLES. 

1  What  is  the  cube  root  of  99252847? 

99,252,847(463  Ans.  463. 

4X4X4=64 


4X4X300=4880 

Div.  4800X6= 
6X6X4X30= 


35252  463 
463 

28800  

4320  1389 


0X6X6=    216          2778 

1852 

Subtrahend       33336  

214369 

46X46X300=634800!  1916847  463 


Div.  634800x3=1904400  643107 

3X3X46X30=     12420  '    1286214 

3X3X3=  27  857476 


Subtrahend        1916847      Proof  99252847 

2  What  is  the  cube  root  of  84604519?       -Ans.  439. 

3  What  is  the  cube  root  of  259694072?       Ans.  638. 

4  What  is  the  cube  root  of  32461759?         Ans.  319. 

5  What  is  the  cube  root  of  5735339?  Ans.  179. 

6  What  is  the  cube  root  of  48228544?         Ans.  364. 

7  What  is  the  cube  root  of  673373097125?  Ans.  87C5. 

8  What  is  the  cube  root  of  7532641?    Ans.  196.02-f- 

9  What  is  the  cube  root  of  5382674.      Ans.  175.2-j- 

10  What  is  the  cube  root  of  15926.972504? 

Ans.  25.16+ 

When  decimals  occur,  point  the  periods  both  ways,  beginning  at  the 
decimal  point,  and  if  the  last  period  of  the  decimal  be  not  complete, 
add  one  or  more  cyphers. 

A  mixed  number  may  be  reduced  to  an  improper 
fraction,  or  a  decimal,  and  the  root  thereof  extracted. 


*  A  surd  is  a  quantity  whose  root  cannot  exactly  be  formed, 
quantity  whose  root  can  be  found,  is  called  a  rational  quantity. 


176  THE   CUBE   ROOT. 

1  What  is  the  cube  root  of  ^VV  ?  Ans. 


. 

2  What  is  the  cube  root  of    |J?  Ans   5-. 

3  What  is  the  cube  root  of          ?  Ans     . 


4  What  is  the  cube  root  of  12UJ  Ans.  2J. 

5  What  is  the  cube  root  of  Sl^s  ?  Ans.  31. 

SURDS. 

6  What  is  the  cube  root  of  7^?  Ans.  1.93-(- 

7  What  is  the  cube  root  of  91?    «          Ans.  2.092+ 

APrLICATION. 

1  The  cube  of  a  certain  number  is  103823;  what  is 
that  number?  Ans.  47 

2  The  cube  of  a  certain  number  is  1728  j  what  num- 
ber is  it?  Ans.  12. 

4  There  is  a  cistern  or  vat  of  a  cubical  form,  which 
contains  1331  cubical  feet:  what  are  the  length,  breadth 
and  depth  of  it?  Ans.  each  11  feet. 

4  A  certain  stone  of  a  cubical  form  contains  474552 
solid  inches  j  what  is  the  superficial  content  of  one  of  its 
sides?  *  Ans.  6084  inches. 


Questions. 

What  is  a  cube?  A  cube  is  a  solid  body  contained  by 
six  equal  square  sides. 

What  is  the  cube  root?  It  is  the  length  of  one  side 
of  a  cube. 

What  is  the  square  of  the  cube  root?  It  is  the  su- 
perficial contents  of  one  side  of  a  cube. 

How  do  you  point  off  a  number  whose  cube  root  is  tc 
be  extracted? 

What  is  the  first  figure  of  the  root?  It  is  the  root  of 
the  greatest  cube  in  the  first  period. 

When  you  subtract  the  cube  from  the  first  period, 
what  do  you  do? 

How  do  you  find  the  divisor? 

What  is  the  first  step  towards  finding  the  subtrahend  ? 
What  is  the  second?  What  is  the  third? 

When  a  remainder  occurs,  how  do  you  proceed? 

How  do  you  prove  the  cube  root? 


THE    CUBE    ROOT. 


177 


Illustration  of  the  Rule  for  extracting  the  Cube  Root. 

The  reason  for  pointing  off  the  number  into  periods 
of  three  figures  eacli,  is  similar  to  the  one  given  in  the 
Square  Root;  for  the  number  of  figures  in  any  cube  will 
never  exceed  three  times  the  figures  in  the  root,  and 
will  never  be  more  than  two  figures  less. 

OPERATION. 

15,625  I  25 

8 


2X2X2= 
2X2X300=1200 


5X5X2X30= 
•       5X5X5= 


7625 

6000 

1500 

125 

7625 


Fig.  1. 


In  Ois  number  there  are 
two  pel  ->ods :  of  course  there 
will  be  two  figures  in  the 
root. 

"The greatest  cube  in  the 
left  hand  period  (15)  is  8, 
the  root  of  which  is  2;" 
therefore,  %  is  the  first  figure 
of  the  root,  and  as  we  shall 
have  another  figure  in  the 
root,  the  2  stands  for  2  tens, 
or  20.  But  the  cube  root  is 
the  length  of  one  of  the  sides 
of  the  cube,  whose  length, 
breadth  and  thickness  are 
equal :  then  the  cube  whose 
root  is  20,  contahw  20X20 
X  20=8000. 

"Subtract  the  cube  thus 
found  (8)  from  said  period, 
and  to  the  remainder  bring 
down  the  next  period,"  or, 
subtract  the  8000  from  the 
whole  given  number  (15625) 
and  7625  will  remain.  Thus 
8000  feet  are  disposed  of  in 
the  cube,  Fit  1.  20ft  .ong, 
80  ft  wide,  and  20  ft.  nigh. 

The  cube  is  to  be  enlarged 
by  the  addition  of  7625  feet 
which  remain.  In  doing 

this,  the  figure  must  be  enlarged  on  three  sides,  to  make  it  longer^ 

and  wider,  and  higher,  to  maintain  the  complete  cubic  form. 

The  next  step  is,  to  find  a  divisor;   and  this  must  oe  the  number  of 

square  feet  contained  in  the  three  sides  to  which  the  addition  must  be 

made. 

Hence  we  ^multiply  the  square  of  the  quotient  Jigure  by  300."* 

That  is,  2  X  2  X  300=1200 :  or  20  X  -0  X  3=  1 200  feet,  which  is  the 

superficial  content  of  the  three  sides,  A,  B,  and  C. 


H  2 


178 


THE    CUBE    ROOT. 


Fig.  2. 


Fig.  4. 


Proof. 

20X20X20= 
20X20X3X5= 
5X5X20X3= 
5X5X5= 


8000 

6000 

1500 

125 


25X25X25=       15625 


This  "divisor  (1200)  is 
contained  in,  ike  dividual" 
(7625)  5  times :  then  5  is  the 
second  quotient  figure ;  that 
is,  the  addition  to  each  of  the 
three  sides  is  5  feet  thick ;  if 
1200  feet  cover  the  three 
sides  one  foot  thick,  5  feet 
thick  will  require  5  times  as 
many;  that  is  1200X5= 
6000. 

But  when  the  additions 
are  made  to  the  three  squares 
there  will  be  a  deficienc)- 
along  the  whole  length  ol 
the  sides  of  the  squares  be- 
tween the  additions,  which 
must  be  supplied  before  the 
cube  will  be  complete.  These 
deficiencies  will  be  three,  as 
may  be  seen  at  NNN  in 
Fig.  2,  therefore  it  is  that 
we  "multiply  the  square  of 
the  last  figure  by  the  prece- 
ding figure,  and  by  30," 
(that  is,  5X5X^X30,)  or 
5  X  5  X  20  X  3=1500vvhich 
is  the  quantity  required  to 
supply  the  three  deficiencies. 

Figure  3,  represents  the 
solid  with  these  deficiencies 
supplied,  and  discovers  an 
other  deficiency,  where  they 
approach  each  other  at  ooo. 

Lastly,  "cube  the  last  fig- 
ure;'1'' this  is  done  to  fill  the 
deficiency  left  at  the  comer, 
in  filling  up  the  other  defi- 
ciencies. This  corner  is 
limited  by  the  three  portions 
applied  to  fill  the  former  va . 
cancies,  which  were  5  feet  in 
breadth ;  consequently  the 
cube  of  5  will  be  the  solid 
contents  of  the  corner.  Fig. 
4  represents  this  deficiency 
(eee)  supplied,  and  the  cube 
complete. 


ROOTS    OP   ALL    POWERS.  179 

The  illustration  is  much  better  made  by  means  of  8  blocks 'of  the 
ollowing  description:  One  cube  of  about  3  inches  diameter;  three 
lieces  each  3  inches  square,  £  inch  thick;  three  pieces  each  &  inch 
square,  3  inches  long ;  and  one  cube  %  inch.  A  set  of  these  should 
>elong  to  the  apparatus  of  every  Professional  Teacher. 

A  GENERAL  RULE    FOR  EXTRACTING  THE 
ROOTS  OF  ALL  POWERS. 

1  Point  the  given  number  into  periods,  agreeably  to 
the  required  root. 

2  Find  the  first  figure  of  the  root  by  the  table  of  pow- 
ers, or  by  trial ;   subtract   its  power   from  the  left  hand 
seriod,  and  to  the  remainder  bring  down  the  first  figure 
in  the  next  period  for  a  dividend. 

3  Involve  the  root  to  the  next  inferior  power  to  that 
which  is  given,  and  multiply  it  by  the  nuhiber  denoting 
the  given  power,  for  a  divisor;   by  which  find  a  second 
figure  of  the  root. 

4  Involve   the  whole   ascertained   root  to  the  given 
power,  and  subtract  it  from  the  first  and  second  periods 
Bring  down  the  first  figure  of  the  next  period  to  the  re- 
mainder, for  a  new  dividend;  to  which,  find  a  new  divi- 
sor, as  before;  and  so  proceed. 

Note.— The  roots  of  the  4th,  6th,  8th,  9th,  and  12th 
powers,  may  be  obtained  more  readily  thus : 

For  the  4th  root  take  the  square  root  of  the  square 
root. 

For  the  6th,  take  the  square  root  of  the  cube  root. 

For  the  8th,  take  the  square  root  of  the  4th  root. 

For  the  9th,  take  the  cube  root  of  the  cube  root. 

For  the  12th,  take  the  cube  root  of  the  4lh  root. 

EXAMPLES. 

1  What  is  the  5th  root  of  916132832? 
9161,32832(62  Ans. 
7776  6X6X6X6X6=7776 

6X6X6X6X5=6480  div 

6480)13853 


916132832  62x62x62x62x62=916132832 
916132832 


180  ARITHMETICAL    PROGRESSION. 

2  What  is  the  fourth  root  of  140283207936?  Ans.  612. 

3  What  is  the  sixth  root  of  782757789696  ?     Ans.  96. 

4  What  is  the  seventh  root  of  194754273881  ?  Ans.  41. 

5  What  is  the  ninth  root  of  1352605460594688  ? 

Ans.  48. 


ARITHMETICAL  PROGRESSION. 

A  SERIES  of  numbers,  increasing  or  decreasing  by  a 
common  difference,  is  called  an  Arithmetical  Progres- 
sion. 

Thus  3,  5,  7,  9,  11,  13,  15,  &c.,  is  an  ascending  se- 
ries, whose  common  difference  is  2. 

And  16,  13,  10,  7,  4,  1,  is  a  descending  series,  whose 
common  difference  is  3. 

The  three  most  important  properties  of  an  arithmeti- 
cal series  are  the  following : 

I.  The  sum  of  the  two  extremes  is  equal  to  twice  the 
mean,  or  to  the  sum  of  any  two  terms  equidistant  from 
the  mean. 

In  the  above  series  3 -{-15=  twice  9,  which  is  the 
mean  or  middle  term;  and  5-f-13  which  are  equidistant 
from  the  mean. 

II.  The  difference  of  the  extremes,  is  equal  to  the 
common  difference  multiplied  by  the  number  of  terms, 
less  one. 

In  the  above,  the  number  of  terms  7 — 1=6;  then 
the  common  difference  2X6  =  12,  which  is  equal  to 
15 — 3.  Or  the  number  of  terms  in  the  other  series 
6—1=5  ;  then  5X3  =  15,  which  is  equal  to  16—1. 

III.  The  sum  of  all  the  terms  is  equal  to  the  product 
of  the  mean,  or  of  half  the  sum  of  the  extremes,  multi- 
plied by  the  number  of  terms. 

As  above,  the  mean  is  9  ;  which,  multiplied  by  7,  the 
number  of  terms,  gives  63= 15-j-13-|-  11 4-9-f  7+5-1-3. 

Or,  15+3  =  18;  half  of  which  is  9.  Then,  9X7 
=63,  as  before. 


ARITHMETICAL    PROGRESSION.  181 

And  16-fl  =  17;  half  of  which  is  8£.  This  multi- 
tiplied  by  6,  the  number  of  terms,  =51  =  16-J-13-{-10 

+7+4+1. 

NOTE  1. — To  find  the  last  term,  multiply  the  common 
difference  by  the  number  of  terms,  less  one,  and  add  the 
product  to  the  first  term  in  an  increasing  series ;  or,  sub- 
tract the  product  from  the  first  term  in  a  decreasing 
series. 

EXAMPLES. 

1  If  the  first  term  is  3,  the  common  difference  2,  and 
the  number  of  terms  7,  what  is  the  last  term  ? 

Ans.  15. 
7 — 1=6  ;  the  2X6+3=15,  the  last  term. 

2  The  first  term  being  16,  the  common  difference  3, 
and  the  number  of  terms  6,  what  is  the  last  term  ? 

Ans.  1. 

6—1=5;  then  3X5  =  15.  Then  16— 15=1,  the  last 
term. 

3  What  is  the  last  term  in  a  series,  whose  first  term 
is  5,  the  common  difference  4,  and  the  number  of  terms 
25?  Ans.  10K 

4  Suppose,  in  the  above,  the  first  term  is  3  ;  what  is 
the  last  term  ?  Ans.  99. 

5  A  man  bought  50  yards  of  calico  at  6  cents  for  the 
first  yard,  9  for  the  second,  12  for  the  third,  &c.;  what 
did  he  pay  for  the  last?  Ans.  SI. 53. 

NOTE  2. —  To  find  the  mean  term,  take  half  the  sum 
of  the  extremes. 

EXAMPLES. 

1  The  first  term  is  3,  and  the  last  15 ;  what  is  the  arith- 
metical mean  ?  Ans.  9. 

3+15=18  ;  the  18-^-2=9,  the  mean  term. 

2  The  weight  of  5  packages  of  goods  is,  severally, 
180,   150,   120,  90,  60  pounds  ;    what  is  the  mean  or 
average  weight  ?  Ans.  1201bs. 

NOTE  3. —  To  find  the  sum  of  all  the  terras,  multiply 
the  mean  term  by  the  number  of  terms  ;  or,  the  mean 


182  ARITHMETICAL    PROGRESSION. 

by  the  sum  of  the  two  extremes,  and  take  half  the  pro- 
duct. 

EXAMPLES. 

1  The  mean  term  is  11,  and  the  number  of  terms  9; 
what  is  the  sum  of  the  series  ?  Ans.  99. 

2  The  first  term  is  5,  the  last  32,  and  the  number  of 
terms  10  ;  what  is  the  sum  of  the  series  ?     Ans.  185. 

3  How  many  strokes  does  the  hammer  of  a  common 
clock  strike  in  12  hours  ?  Ans.  78. 

4  What  debt  can  be  discharged  in  one  year,  by  week- 
ly payments  in  arithmetical  progression,  the  first  being 
$12,  and  the  last,  or  fifty-second,  payment  $1236  ? 

Ans.  32448. 

NOTE  4. — To  find  the  common  difference,  divide  the 
difference  of  the  extremes  by  the  number  of  terms,  less 
one. 

EXAMPLES. 

1  The  ages  of  8  boys  form  an  arithmetical  series — 
the  youngest  is  4  years  old  and  the  oldest  is  18  pwhat 
is  the  common  difference  ?  Ans.  2. 

2  A  debt  can  be  discharged  in  one  year,  by  weekly 
payments  in  arithmetical  progression — the  first  is  $12, 
and  the  last  $1236  ;  what  is  the  common  difference  ? 

Ans.  $24. 

NOTE  5. — To  find  the  number  of  terms,  divide  the 
difference  of  the  extremes  by  the  common  difference,  and 
add  1  to  the  quotient. 

EXAMPLES. 

1  In  a  series,  whose  extremes  are  4  and  1000,  and  the 
common  difference  12,  what  is  the  number  of  terms  ? 

Ans.  84. 

2  If  a  man,  on  a  journey,  travels  18  miles  the  fir^t 
day,  increasing  the  distance  2  miles  each  day,  and  on  the 
last  day  goes  48  miles,  how  many  days  did  he  travel  ? 

/  Ans.  16. 


ARITHMETICAL  PROGRESSION.  183 

PROMISCUOUS    EXERCISES. 

1  24  persons  bestowed  charity  to  a  beggar — the  first 
gave  him  12  cents,  the  second  18,  &c.,  in  an  aiithmeti- 
cal  series  ;  what  sum  did  he  receive  ?        Ans.  $19.44. 

2  Suppose  100  apples  were  placed  in  a  right  line,  2 
yards  apart,  and  a  basket  2  yards  from  the  first ;  how 
far  would  a  boy  travel  to  gather  them  up  singly,  and  re- 
turn with  each  separately  to  the  basket  ? 

Ans.  20200  yards. 

3  In  a  drove  of  400  hogs,  5  of  the  largest  weighs,  on 
an  average,  280  pounds  a-piece,  and  5  of  the  smallest 
180 ;  what  is  the  mean  Or  average  weight  of  them  all, 
and  what  is  the  whole  weight  ? 

.        C  2301bs.  mean  weight. 
S*  I    92000  Ibs.  whole  wt. 

4  How  many  acres  in  a  piece  of  land  80  rods  wide 
at  one  end  and  60  at  the  other,  and  1 20  rods  long  ? 

Ans.  52£. 

It  may  be  observed,  that  the  natural  numbers  1,  2,  3, 
4,  5,  6,  7,  &c.,  is  an  arithmetical  series,  whose  first  term 
is  1,  and  common  difference  1 ;  and  that  the  last  term 
is  equal  to  the  number  of  terms. 

From  this  series,  we  may  form  another  by  adding  to 
each  figure  the  sum  of  all  the  preceding,  and  we  shall 
have  1,  3,  6,  10,  15,21,28,  &c. 

These  are  called  triangular  numbers,  because  they 
may  be  represented  by  points,  forming  equilateral  trian- 
gles, thus : 


Hence  we  perceive,  that  the  sum  of  the  natural  numbers, 
to  any  degree,  expresses  the  triangular  number  of  the 
same  degree. 

In  the  same  manner,  the  square  numbers  1,  4,  9,  16, 


184  GEOMETRICAL    PROGRESSION. 

25,  36,  &c.,  may  be  expressed  by  points,  arranged  in 
squares,  thus : 


Natural  numbers— -1,  2,  3,  4,  5,  6,  7,  8,  9,  &c. 
Triangular  numbers— 1,  3,  6,  10,  15,  21,  28,  36,  &c. 
Square  numbers— 1,  4,  9,  16,  25,  36,  49,  64,  81,  <fcc. 
Cube   numbers— 1,   8,  27,  64,   125,  216,  343,  512, 
729,  &c. 


GEOMETRICAL  PROGRESSION. 

A  SERIES  of  numbers,  increasing  or  decreasing  by  a 
common  ratio,  is  called  a  Geometrical  Progression. 

Thus  2,  4,  8,  16,  32,  64,  128,  is  an  increasing  series 
whose  common  ratio  is  2  ; 

And  729,  243,  81,  27,  9,  3,  is  a  decreasing  series, 
whose  common  ratio  is  £. 

The  most  important  properties  of  a  geometrical  series 
are  the  following : 

I.  The  product  of  the  extremes  is  equal  to  the  square 
of  the  mean ;  or,  to  the  product  of  any  two  terms  equi 
distant  from  the  mean. 

In  the  above,  128X2  =  16X  16,  or  32X8,  &c.  Also 
729X3=243X9,  or  81X27,  between  which  the  mean 
falls. 

Hence,  the  mean  term,  in  a  geometrical  series,  is  th 
square  root  of  the  product  of  the  extremes,  or  of  an) 
two  terms  equidistant  from  the  mean. 

II.  The  last  term  of  an  increasing  series,  is  the  pro 
duct  of  the  first  term,  multiplied  by  the  ratio  involved  t( 
the  power  which  is  one  less  than  the  number  of  terms 
in  a  decreasing  series,  it  is  the  quotient  of  the  first  term 
divided  by  the  power. 

In  an  increasing  series,  whose  first  term  is  2,  ratio  3 


GEOMETRICAL    PROGRESSION.  185 

and  the  number  of  terras  5,  we  have,  by  the  natural 
method,  2,  6,  18,  54,  162,  which  gives  162  for  the  last 
term. 

Or,  by  the  artificial  method,  the  ratio  involved  in  the 
4th  power,  which  is  one  less  than  the  number  of  terms, 
3X3X3X3=81;  these,  multiplied  by  the  first  term, 
2X81,  gives  162,  the  last  term,  as  before. 

Again,  let  the  first  term  be  4 ;  then  we  have  4,  12,  36, 
108,  324,  or  3X3X3X3  =  81.  Then  multiply  the  first 
term,  4,  by  81=324,  the  last  term. 

In  a  decreasing  series,  with  the  first  term  243,  ratio  £, 
and  the  number  of  terms  5,  we  have  243,  81,  27,  9,  3 ; 
then  3X3X3X3=81,  and  243-f-81  gives  3,  which  is 
the  last  term,  as  before. 

NOTE  1. — To  find  the  last  term,  involve  the  ratio  to 
the  power  which  is  one  less  than  the  number  of  terms, 
and  multiply  the  first  term  by  the  power. 

EXAMPLES. 

1  What  is  the  eighth  term  of  an  increasing  geometri- 
cal series,  whose  first  term  is  4,  and  ratio  2  ? 

Ans.  512. 

2X2X2X2X2X2X2  =  128;  then  4X128  =  512, 
which  is  the  eighth  term.  Or,  4,  8,  16,  32,  64,  128,  256, 
512,  the  eighth  term,  as  before. 

2  Required  the  last  term  of  an  increasing  series,  whose 
first  term  is  15,  ratio  2,  and  number  of  terms  10. 

Ans.  7680. 

3  A  boy  purchased  18  oranges,  at  1  cent  for  the  first, 
4  for  the  second,  16  for  the  third,  &c.;  what  was  the 
price  of  the  last?  Ans.  $171798691.84. 

NOTE  2. — To  find  the  sum  of  all  the  terms,  multiply 
I  the  last  term  by  the  ratio,  and  from  the  product  subtract 
the  first  term  ;  then  divide  the  remainder  by  the  ratio,  less 
one. 

EXAMPLES. 

1  What  is  the  sum  of  a  series,  whose  first  term  is  2, 
ratio  3,  and  number  of  terms  5  ?  Ans.  242. 


186  GEOMETRICAL   PROGRESSION. 

Find  (he  last  term  by  Note  1 . 

162  last  term. 
2  3  ratio. 


486 

2  first  term. 


3—1=2)484 
242  

242  sum  of  the  sums. 

2  What  is  the  sum  of  the  series,  whose  first  term  ii 
4,  ratio  5,  and  number  of  terms  7  ?  Ans.  15624. 

Find  the  last  term  by  Note  1. 

12500  last  term. 
4  5  ratio. 

20  

100  62500 

500  4  first  term. 

2500  

12500          5—1=4)62496 

15624  15624  sum  of  the  series. 

3  What  is  the  sum  of  a  series,  whose  first  term  is  3 
ratio  4,  and  number  of  terms  7  ?  Ans.  16383. 

Last  term, 

Ratio, 


First  term, •  . 

4—1=3)49149 


Sum  of  the  series 16383 

Write  down  the  series,  multiply  it  by  the  ratio,  an< 
subtract  the  first  series  from  the  second,  thus  : 

3  12  192  768  3072  12288 

12  192  768  3072  12288  49152 


GEOMETRICAL    PROGRESSION.  187 

Hero  the  terms  all  cancel,  but  the  first  of  the  upper  and 
last  of  the  lower  series.     Then  we  have 

49152 
3 


Divide  by  ratio,  less  1  :     4 — 1=3)49149 

The  sum  of  the  series, 16383 

Now,  as  we  multiplied  the  given  series  by  the  ratio, 
which  is  4,  and  subtracted  once  the  series  from  the  pro- 
duct, the  remainder  is  three  times  the  given  series.  We, 
therefore,  divide  by  3,  which  is  the  ratio,  less  1  :  the  quo- 
tient is  the  sum  of  the  series. 

4  At  2  cents  for  the  first  *ounce,  6  for  the  second,  18 
for  the  third,  &c.,  what  would  a  pound  of  gold  cost  ? 

Ans.  $5314.40. 

5  Sold  10  yard3  of  velvet,  at  4  mills  for  the  first  yard, 
20  for  the  second,  100  for  the  third,  &c.;  what  did  the 
piece  cost  ?  Ans.  $9765.62,4. 

6  What  is  the  cost  of  a  coat  with  14  buttons,  at  5  mills 
for  the  first,  15  for  the  second,  45  for  the  third,  &c.? 

Ans.  $11957.42. 

7  What  is  the  cost  of  16  yards  of  cloth,  at  3  cents  for 
the  first  yard,  12  for  the  second,  48  for  the  third,  &c.? 

Ans.  $42949672.95.- 

NOTE  3. — The  sum  of  a  geometrical  series  is  found 
by  the  extremes  and  the  ratio,  independent  of  the  num- 
ber of  terms ;  hence,  whether  the  number  of  terms  be 
many  or  few,  there  is  no  variation  in  the  rule.  We  may, 
therefore,  require  the  sum  of  the  seres,  6,  3,  1, 1,  -*-,  &c., 
to  infinity,  provided  we  can  determine  the  value  of  the 
other  extreme.  Now,  we  see  the  terms  decrease  as  the 
series  advances  ;  and  the  hundredth  term,  for  example, 
would  be  exceedingly  small,  the  thousandth  too  small  to 
be  estimated,  the  millionth  still  less,  and  the  infinite  term 
would  be  nothing :  not,  as  some  tell  us,  "  extremely 
small,"  or,  "  too  little  to  be  considered,"  &c.,  but  abso- 
lutely nothing. 


188  GEOMETRICAL    PROGRESSION. 

Now  let  us  consider  the  series  as  inverted ;  then  6  will 
be  the  last  term,  and  3  the  ratio.  By  the  rule,  multiply 
the  last  term  by  the  ratio,  subtract  the  first  term,  and 
divide  by  the  ratio,  less  1.  Here  we  have  the  sum  c€ 

6X3—0 
the  series, =9,  the  answer. 


EXAMPLES. 

1  What  is  the  sum  of   the  infinite   series,  1-j-i-f-j 
-f-yV'  &c.?     Invert  the  series  :  £  is  the  last  term,  and  2 

£X  10—0 
the  ratio;   hence,  -  =1,  the  answer. 

2  What  is   the  sum  of  the  infinite  series   y\,  Tf^, 

_3_xiO—  0 
y^,  &c.?     --  -  -  =  |,  the  answer. 

9 

3  What  is  the  value  of  \,  ~,  TV  j,  &c.,  to  infinity  ? 

Ans.  |. 

4  What  is  the  value  of  |,  5,  i,  f,  &c.,  to  infinity? 

Ans.  f. 

5  What  is  the  value  of  1,  |,  T9g,  &c.,  to  infinity  ? 

Ans.  4. 
Here  the  ratio  is  |. 

6  What  is  the  value  of  f  ,  54T,  T§j,  &c.,  to  infinity  '• 

Ans.  |. 

7  What  is  the  value  of  .777,  &c.,  to  infinity  ?     This 
may  be  expressed  by  T\,  TJT,  T¥\o  »  &c- 


8  What  is  the  sum  of  .6666,  &c.,  to  infinity  ? 


Ans.  •. 


9  What  is  the  value  of  .232323,  &c.,  infinitely  ex- 
tended ?  Ans.  ff 

This  may  be  expressed  by  T2/o»  Tf  lo»  &c- 


EXCHANGE.  .        189 \ 

EXCHANGE. 

THE  object  of  exchange  is  to  find  how  much  of  the 
money  of  one  country  is  equivalent  to  a  given  sum  of  the 
money  of  another. 

By  the  par  of  exchange  between  two  countries,  is 
meant  the  intrinsic  value  of  the  one,  compared  with  the 
other ;  it  is  estimated  by  the  weight  and  fineness  of  the 
coins. 

The  course  of  exchange,  at  any  time,  is  the  sum  of 
the  money  of  one  country  which,  at  that  time,  is  given 
for  a  certain  sum  of  money  of  another  country.  The 
course  of  exchange  varies  according  to  the  circumstances 
of  trade.  All  the  calculations  in  exchange  can  be  per- 
formed by  the  Rule  of  Three. 

EXAMPLES. 

1  A  sovereign,  of  England,  is  worth  $4.84,6.    A  mer- 
chant of  New  York  is  indebted  to  a  merchant  of  London 
$7520 ;  how  many  sovereigns  will  it  require  to  discharge 
the  debt  ? 

2  A  six-ducat  piece  of  Naples,  is  wprth  $5.25.     A 
merchant  of  Naples  is  indebted  to  a  merchant  in  Boston 
$6940  ;  how  many  such  pieces  shall  he  remit  ? 

3  The  current  rupee  of  Calcutta  is   44.4   cents.     A 
merchant  of   Philadelphia  has   a  claim  against  a  mer- 
chant of  Calcutta  of  $437  ;  how  many  rupees  shall  he 
draw  ? 

4  The  piaster  is  20£  cents  ;    how  many  piasters  in 

$1128.22? 

5  Reduce  7218  rupees  to  Federal  money,  at  46  cents 
per  rupee. 

6  The  dollar  of  Bencoolen  is  reckoned  at  $1.10,  Fed- 
eral  money ;   how  much   Federal  money  in  $2740  of 
Bencoolen  ? 

7  The  Prussian  rix-dollar  is  worth  66|  cents.     Re- 
duce $7348.32  into  Prussian  money. 


190  PROMISCUOUS    EXERCISES. 

Ex.  1.  If  l£  sterling  is  worth  $4,44  cts.  4  ms. ;  what 
is  65^6  sterling  worth  ?  Ans.  $288,86. 

2.  What  is  the  value  of  $500  in  English  money,  at 
$4,44,4,  per  £  sterling?  Ans.  112£°  10s.  2£<L 

3.  What  is  the  value  of  125^  7s.  at  $4,44,4,  per  £ 
sterling  ?  Ans.  $557,05,5. 

4.  What  is  the  value  of  $1000,  in  English  money,  at 
$4,44,4,  per  £  sterling  Ans.  225^  5*d. 


PROMISCUOUS  EXERCISES. 
1.  A  merchant  had  1000  dollars  in  bank ;  he  drew  out 
at  one  time  $237.50,  at  another  time,  $116.09,  and  at 
another,  $241.061 :  after  which  he  deposited  at  one  time 
1500  dollars,  and  at  another  time  $750.50 ;  how  much 
had  he  in  bank  after  making  the  last  deposite  ? 

Ans.  $2655.84£. 

2  Sold  8  bales  of  linen,  4  of  which  contained  9  pieces 
each,  and  in  each  piece  was  35  yards ;  the  other  4  bales 
contained   12   pieces  each,   and   in  each   piece  was  27 
yards ;   how  many  pieces  and  how  many  yards  were  in 
all  ?  Ans.  84  pieces,  2556  yards. 

3  If  a  man  leave  6509  dollars  to  his  wife  and  two  sons, 
thus — to  his  wife  f ,  to  his  elder  son  |  of  the  remainder, 
and  to  his  other  son  the  rest ;  what  is  the  share  of  each  ? 

("Wife's  share  $2440.87 1. 
Ans.  1  Elder  son's     $2440.87^. 
[Other  son's    $1627.25. 

4  What  is   the  commission  on  $2176.50,  at  2£  per 
cent?  Ans.  $54.411. 

5  If  a  tower  is  384  feet  high  from  the  foundation,  a 
sixth  part  of  which  is  under  the  earth,  and  an  eighth 
part  under  water,  how  much  is  visible  above  the  water  ? 

Ans.  272  feet. 

6  How  many  bricks  9  inches  long  and  4  inches  wide, 
will  pave  a  yard  that  is  20  feet  square  ?          Ans.  1600. 

7  What  is  the  value  of  a  slab  of  marble,  the  length  of 
which  is   5  feet  7  inches,  and  the  breadth  1  foot  10 
inches,  at  1  dollar  per  foot?  Ans.  $10.232. 

8  A  certain  stone   measures  4  feet  6  inches  in  length, 


PROMISCUOUS    EXERCISES.  191 

2  feet  9  inches  in  breadth,  and  3  feet  4  inches  in  depth; 
how  many  solid  feet  does  it  contain?       Ans.  41  ft.  3  in. 

9  A  line  35  yards  long  will  exactly  reach  from  the 
top  of  a  fort,  standing  on  the  brink  of  a  river,  to  the  op- 
posite bank,  known  to  be  27  yards  from  the  foot  of  the 
wall;  what  is  the  height  of  the  wall? 

Ans.  22  yards  3\  inches.-f- 

10  The  account  of  a  certain  school  is  as  follows,  viz. 
~  of  the  boys  learn  geometry,  f  learn  grammar,  ~ 
learn  arithmetic,  —o  learn  to  write,  and  9  learn  to  read: 
what  number  is  there  of  each? 

A          (5  who  learn  geometry,  30  grammar,  24 
I     arithmetic,  12  writing,  and  9  reading. 

11  A  merchant,  in  bartering  with  a  farmer  for  wood 
at  $5  per  cord,  rated  his  molasses  at  $25  per  hhd.,  which 
was  worth  no  more  than  $'20 ;  what  price  ought  the  far- 
mer to  have  asked  for  his  wood  to  be  equal  to  the  mer- 
chant's bartering  price?  Ans.  $6,25. 

12  A  and  B  dissolve  partnership,  and  equally  divide 
their  gain :     A's  share,  which  was  $332  50  cts.,  lay  for 
21  months;  B's  for  9  months  only:  the  adventure  of  B. 
is  required.  Ans.  $775  834  cts. 

13  If  a  water-hogshead  holds  110  gals,  and  the  pipe 
which  fills  the  hogshead  discharges  15  gal.  in  3  minutes, 
and  the  tap  will  discharge  20  gal.  in  5  minutes,  and  these 
were  both  left  running  one  hour,  how  many  gallons 
would  the  hogshead  then  contain;   and  if  the  tap  was 
then  stopped,  in  what  time  would  the  hogshead  be  rilled? 

Ans.  60  gal.,  and  filled  in  10  min. 

14  A  has  B's  note  for  $500  75  cts.,  with  9  months  in- 
terest, at  6  per  cent.,  v^e  on  it,  for  which  B  gave  him 
5064  feet  of  boards,  a>  <\  ,*,ts  per  foot,  with  140  pounds 
of  tallow,  at  13  cts.  pei  n^Mid,  and  is  to  pay  the  rest  in 
flax  seed,  at  92$  cts.  per  b.Vnljhow  many  bushels  of  flax 
seed  must  A  receive,  to  ba,A\ce  the  note? 

Ans.  409Jf  £.  bushels. 

15  A,  B,  an3  C,in  company,  had  put  in  $5762:   A's 
money  was  in  5  months,  B's  7,  and  C's  9  months:  they 
gained  $780,  which  was  so  divided,  that  1  of  A's  was  J 
of  B'SJ  and  j  of  B's  was  1  of  C's :  but  B,  having  Deceived 


192  PROMISCUOUS    EXERCISES. 

$•2087,  absconded:  what  did  each  gain,  and  put  in;  and 
what  did  A  and  C  gain  or  lose  hv  B's  misconduct? 
f  A's  stock  $2494,887         gain  260 
A       J  B's     do     $2227,577  clo    325 

"  ]  C's     do     $1039,536          do    195 
I^A  and  C  would  gain          $465,577 

16  When  100  boxes  of  prunes  cost  2  dollars  10  cents 
each,  and  by  selling  them  at  3  dollars  50  cents  per  cwt. 
the  gain  is  25  percent.,  the  weight  of  each  box,  one  with 
another,  is  required.  Ans.  84  Ib. 

17  There  are  two  columns,  in  the  ruins  of  Persepo- 
iis,  left  standing  upright;  one  is  64  feet  above  the  plain, 
the  other  50.     Between  these,  in  a  right  line  stands  an 
ancient  statue,  the  head  whereof  is  97  feet  from  the  sum- 
mit of  the  higher,  and  86  feet  from  the  top  of  the  lower 
column,  and  the  distance  between  the  lower  column  and 
the  centre  of  the  statue's  base,  is  76  feet;  the  distance 
between  the  top  of  the  columns  is  required.  Ans.  157-J-ft 

18  If  I  see  the  flash  of  a  cannon,  fired  from  a  fort  on 
the  other  side  of  a  river,  and  hear  the  report  47  seconds 
afterwards,  what  distance  was  the  fort  from  where  I 
stood?  Ans.  53674  feet. 

NOTE. — Sound,  if  not  interrupted,  will  move  at  the  rate  of  about 
1142  feet  in  a  second  of  time. 

19  What   is  the  difference  between  the  interest  of 
$1000  at  6  per  cent,  for  8  years,  and  the  discount  of  the 
same  sum  at  the  same  rate,  and  for  the  same  time  ? 

Ans.  The  interest  exceeds  the  discount  by 
$155  67  cts.  5  m. 

20  If  a  tower  be  built  in  the  following  manner,  7\\ 
of  its  height  of  stone,  27  feet  of  brick,  and  i  of  its  height 
of  wood,  what  was  the  height  of  the  tower? 

Ans.  113  feet  4  inches. 

21  A  captain,  2  lieutenants,  and  30  seamen,  take  a 
prize  worth  $7002,   which  they  divide  into  100  shares, 
of  which  the  captain  takes  12,  the  two  lieutenants  each 
5,  and  the  remainder  is  to  be  divided  equally  among  the 
sailors;  how  much  will  each  man  receive?     Ans.  Cap- 
tain's share,  $840.24,   each  lieutenant's,  $350,10,  and 
each  seaman's,  $182,05,2. 


PREFACE  TO  APPENDIX. 


THE  author  of  the  foregoing  work,  has  long  contemplated  its  ex- 
Jgnsion  in  an  Ajjoejidix,  which  is  now  offered  to  the  public  in  hopes  | 
the  usefulness  of  the  whole  may  be  extended,  and  the  science  of; 
arithmetic  advanced.  The  view  of  numbers,  and  the  abridged 
modes  of  operation  herein  presented,  it  is  believed,  will  be  found  ac- 
ceptable alike  to  the  business-man  and  to  the  scholar. 

The  Appendix  recognizes  the  scientific  character  of  numbers, 
and  gives  bold  and  enlarged  views  of_arithmetical  operations.  The 
method  of  cancelling  is  not  new^  but_for  a  long  period  it  has 
scarcely  been  known! ITls,  however,  coming  into  general  use; 
and  it  is  carried  much  farther  in  this  part  of  the  work  than  in  any 
we  have  hitherto  known. 

In  Europe,  this  system  has  been  very  generally  adopted  in  the 
higher  schools,  and  in  this  country  it  is  fast  becoming  known — and 
as  far  as  it  is  known,  it  supercedes  the  usual  modes  of  operation. 

To  the  method  of  stating  problems  in  Proportion  by  comparing 
cause  and  effect,  we  invite  special  attention.  On  critical  examina- 
tion it  will  be  found  more  easy  and  more  rational  than  any  other 
method. 

Other  methods  of  statement  sometimes  require  the  number  of 
men  to  be  multiplied  into  feet  of  wall — days  into  acres  of  grass,  &c., 
all  of  which,  though  correct  as  abstract  proportion  in  numbers,  is 
unnatural  and  void  of  strict  philosophical  expression — not  so  witt 
this  method. 

The  peculiarities  which  the  student  will  here  find  in  the  Extrac- 
tion of  Roots,  and  in  Mensuration,  are  not  all  new — indeed  there 
an  be  nothing  new  in  principle — but  as  far  as  the  author's  know- 
ledge extends,  he  is  not  aware  that  these  abbreviations  have  ever 
been  collected  in  any  arithmetical  work.  The  impression  seems  to 
have  been,  that  the  people  could  not  comprehend  arithmetical  breV' 
ity,  nor  appreciate  mathematical  beauty";  but  the  author  thinking 
otherwise,  presents  this  brief,  yet  comprehensive,  Appendix  to  the 
public,  in  the  full  assurance  that  whoever  will  pay  due  attention  to 
the  subject  will  be  highly  gratified  and  abundantly  rewarded. 


I 


193 


APPENDIX. 


THE  pupil  having  passed  over  the  common  routine  of 
Arithmetic,  and  supposed  to  be  able  to  perform  all  its  ne^ 
cessary  operations  in  the  usual  way,  we  now  present  nim 
with  some  new  modes  of  practical  operations,  by  which 
the  labors  will  be  much  abridged  and  the  science  pre- 
sented with  more  of  its  roses  and  less  of  its  thorns. 

We  commence  with  a  systematic  study  of  numbers. 
The  following  are  called  prime  numbers ;  because  no 
one  can  be  divided  by  any  number  less  than  itself  with- 
out producing  a  fraction.  123. 5. 7. ..11.  13... 

17  .  19  ...  23  ..  .  .  29  .  31 37  ...  41  .  43  . 

.  .  47 53  .  .  .  57  .  59  .  61  .  .  .  .  67  .  .  .  71  . 

73 79  ...  83  ...  87  .  89 97  ...  101 

The  points  represent  the  composite  numbers;  and 
here  it  can  be  observed  that  there  are  29  prime  numbers, 
and  of  course  71  composite  numbers  in  the  first  hun- 
dred— the  prime  numbers  becoming  fewer  as  the  num- 
bers rise  higher.  Observe  the  following  series : 

5     10     15     20     25     30     35     40     45     50, 

and  so  on.  Every  body  knows  that  our  Geometric  scale 
of  numbers  i*  1  10  100  1000,  &c.  Now  we  wish  the 
student  to  observe  the  numbers, 

5     20     25     50     75     125     500, 

as  being  not  only  in  the  preceding  series,  but  aliquot 
parts  of  some  number  in  our  Geometric  scale.  For  ex- 
ample, 25  is  I  of  100 ;  125  is  1-  of  1000,  &c. 

We  now  charge  the  student  to  make  his  eye  familiar 
with  all  the  preceding  series — the  prime  numbers  as  be- 
ing unmanageable  and  inconvenient,  and  the  others  the 
very  reverse  ;  but  the  full  importance  of  such  a  study 
can  only  appear  in  the  sequel. 

194 


APPENDIX.  195 

By  a  little  attention  to  the  relation  of  numbers,  we 
may  often  contract  operations  in  multiplication.  A  dead 
unifomity  of  operation  in  all  cases  indicates  a  mechani- 
cal and  not  a  scientific  knowledge  of  numbers.  As  a 
uniform  principle,  it  is  much  easier  to  multiply  by  the 
small  numbers  2,  3,  4,  5,  than  by  7,  8,  9. 

EXAMPLES. 

^   Multiply 4532 

by 39 

Commence    with    the  3    tens.      Mul-  13596 
tiply  this  13596  by  3,  because  3X3=9,  40788 

and  place  the  product  in   the   place   of  

units.  176748 

Multiply  •  •        576  Multiply  this  last  number, 

by.  ...        186  3456,  (which  is  6  times  576) 

by  3,  and  place  the  product 

(6X3=18.)     3456  in  the  place  of  tens,  and  we 

10368  have   180   times   576.      Ob- 

serve  the  same  principle  in 

107136  the  following  examples. 

Multiply  ....        576         Multiply  •  •        40788 
by 618  by ....  497 

Commence  with  6.  3456  (7X7=49.)     285516 

(6X3  =  18.)        10368  1998612 

355968  20271636 

Observe,  that  in  this  last  example  497  is  3  less  than 
500 ;  500  is  \  of  1000,  thefore 

2)40788*000 

20394     000 
Subtract  (3X40788=122364.)  122     364 


20271     636 


196  APPENDIX. 

Multiply  ...........  785460 

by  .............  14412 

First  multiply  by  12,  then  that  9425520 

product  by  "2.  113106240 

11320049520 


Multiply  86416  by  135.     Observe,  135  is.  125+10  - 
125  is  |  of  1000,  therefore 

8)8  64  1  6'0  0  0 


10802  000 
864  160 

Product 11666160 


tg  are  from  Ray's  Arithmetic,  page  34  : 

Multiply  1646  by  365.  As  the  first  factor  is  even, 
and  the  last  ends  in  5,  we  mentally  half  the  one  and 
double  the  other,  which  will  not  affect  the  product,  but 
very  much  contract  the  operation,  we  then  have 

823 
730 


24690 
5761 

600790 

We  can  do  the  same  in  all  such  cases. 

Multiply  999  by  777. 

777000 
Subtract 777 


Product 776223 


APPENDIX.  197 


Multiply.  •          61524 
by.  ...  7209 


5537 1 6     Multiply  this  product  of  9  by 
4429728       8,  because  9  times  8  are  72,  and 

place  the  product  in  the  place 

Product    443646516  of  100,  because  it  is  7200. 

Multiply 1243 

by 636 

7458         First  by  600. 
44748     Multiply  7458  by  6. 

Product 790548 

Multiply  624  by  85.  The  product  will  be  the  same 
as  the  half  of  624  by  the  double  of  85— that  is,  312  by 
170,  or  3120  by  17=53040.  We  do  not  say  that  these 
changes  give  any  advantage  in  this  particular  example; 
we  only  make  them  to  call  out  thought  and  attention 
from  the  pupil. 

Multiply 

by- 


This  may  be  done  by  commencing  with  the  2  ;  then 
that  product  by  2  and  3.  Or  we  may  commence  with 
the  6  units,  and  then  that  product  by  4  ;  because  4  times 
6  are  24. 

Multiply  4386,  or  any  other  number,  by  49.  We 
may  do  this  by  taking  the  number  50  times,  or  ^  of  100 
times,  and  subtracting  once  the  number. 

Multiply  87742  by  65.  This  may  be  done  by  taking 
the  number  1  (100,)  +10+!  °C(10)  times-  That  is 

4387100 
877420 
438710 


Product  •  •    ..5703230 
77* 


198  APPENDIX. 

Multiply  92636  by  150.  It  will  be  much  more  easy 
to  multiply  the  half  of  it  by  300,  which  will  give  the 
same  result. 

Multiply  679  by  279.  Multiply  first  by  9,  and  that 
product  by  3,  put  in  the  place  of  10. 

Multiply  87603  by  9865.  By  the  common  formal 
rule  this  would  be  a  tedious  operation  ;  but  let  us  observe 
that  9865  is  10000 — 135,  135  is  125-f-10 ;  125  is  }  of 
1000.  Therefore, 

876030000                    Subtract  I,  8)87603000 
11826405  


: 10950375 

864203595  Product.  10  times,       776030 


11826405 

Multiply  818327  by  9874.     But  9874  is  10000  less 
126.     Therefore 

8183270000           Less  1  of  818327000 
Subtract     103108202  


102290875 

8080161798  Product.  818327 


103109202 

Multiply  188  by  135,  and  that  product  by  15.  In- 
stead of  doing  this  literally  and  mechanically  according  to 
rule,  we  may  half  188  twice,  and  double  each  of  the  fac- 
tors that  end  in  5,  and  we  shall  have  47  .  270  .  .  30  ;  or, 

47  ' 
8100 


4700 
376 

380700 

How  far  will  a  ship  sail  in   365   days,  at  the  rate 


APPENDIX.  199 

of  8  miles  per  hour?     Here   365X24X8—730X96; 
or, 

73000  less  730X4=2920 
2920 

Product 70080 

EXERCISES    FOR   PRACTICE. 

299 X 299 X299=what  number?  Ans.  26730899 
999X999X999= what  number?  Ans.  997002999 
4962  X  98 = what  number  ?  Ans.  487276 

Multiply  8340745  by  64324.  Observe,  that  32  is  8 
times  4,  and  64  is  2  times  32. 

Multiply  8340745  by  64432.  In  this  last  example, 
commence  at  the  4  in  the  place  of  hundreds. 

Multiply  24  X  25  X  12  X  5,  together.  Here  it  would  be 
no  index  of  even  a  decent  knowledge  of  numbers,  to  obey 
literally  and  in  the  order  the  numbers  are  given ;  yet  how 
many  even  at  the  present  day  would  do  so ! 

Take  the  factor  4,  out  of  the  24,  and  multiply  it  into 
the  25  ;  also,  the  factor  2,  out  of  12,  and  put  it  with  5, 
which  can  be  done  without  effort.  We  then  have 
6X  100  X  10  X  6=36000,  the  answer. 


SECTION  II. 

WE  shall  say  nothing  of  division  in  whole  numbers, 
as  nothing  new  or  interesting  can  be  offered  on  that 
topic  ;  but  we  cannot  forbear  making  a  few  comments 
on  division  in  decimals. 

To  divide,  to  cut  into  parts,  will  not  at  all  times  give 
a  clear  understanding  of  the  operation,  and  confusion 
frequently  arises  from  taking  this  view  of  the  subject; 
we  better  consider  it  as  one  number  measuring  another. 
For  example :  how  often  will  .5  of  a  foot  measure  12 
feet  ?  In  other  words,  divide  12  by  .5  ;  or,  divide  12  by 


200  APPENDIX. 

t.  Here,  if  the  student  should  imagine  that  12  must  be 
cut  into  parts,  he  would  make  a  great  error.  He  must 
divide  120  tenths  into  parts  ;  in  this  case,  into  5  parts — 
because  the  5  is  .5 :  or  he  may  consider  that  \  of  a  foot 
may  be  laid  down  in  12  feet;  that  is,  measure  12  feet  24 
times.  Or  he  may  reduce  the  12  feet  to  half  feet,  and 
then  divide  by  1.  In  all  cases,  the  divisor  and  dividend 
must  be  of  the  same  denomination  before  the  division 
can  be  effected.  But  in  decimals,  these  reductions  are 
made  so  easily,  that  a  thoughtless  operator  rarely  per- 
ceives them ;  hence  the  difficulty  in  ascertaining  the 
value  of  the  quotient. 

We  now  give  a  few  examples,  for  the  purpose  of 
teaching  the  pupil  how  to  use  his  judgment;  he  will 
then  have  learned  a  rule  more  valuable  than  all  others. 

EXAMPLES. 

Divide  15.34  by  2.7.  Here  we  consider  the  whole 
number,  15,  is  to  be  divided  by  less  than  3 ;  the  quo- 
tient must,  therefore,  be  a  little  over  5.  One  figure  then, 
in  the  quotient,  will  be  whole  numbers,  the  rest  decimals. 

Divide  15.34  by  .27.  Here  we  perceive  that  15  is  to 
be  divided,  or  rather  measured,  by  less  then  %  of  1 ; 
therefore  the  quotient  must  be  more  than  3  times  15.  Or 
we  may  multiply  both  dividend  and  divisor  by  100, 
which  will  not  effect  the  quotient,  and  then  we  shall 
have  1534,  to  ba  divided  by  27.  Now  no  one  can  mis- 
take how  much  of  the  quotient  will  be  whole  numbers  : 
the  rest,  of  course,  decimals. 

Divide  45.30  by  .015.  Conceive  both  numbers  to  be 
multiplied  by  1000  ;  then  the  requirement  will-be  to  di- 
vide 45300  by  15,  a  common  example  in  whole  num- 
bers. 

By  attention  to  this  operation,  the  student  will  have 
no  difficulty  in  any  case  where  the  divisor  is  less  than 
the  dividend. 

Here  is  one  of  the  most  difficult  cases : 

Divide  .003753  by  625.5.  In  all  such  examples  as 
this,  we  insist  on  the  formality  of  placing  a  cipher  in 


APPENDIX.  201 

the  dividend,  to  represent  the  place  of  whole  numbers, 
thus : 

625.5)0.0037530( 

We  now  consider  whether  the  whole  number  in  the 
divisor  will  be  contained  in  the  whole  number  in  the  div- 
idend, and  we  find  it  will  not ;  wr,  therefore,  write  a 
cipher  in  the  quotient  to  represent  the  place  of  whole 
numbers,  and  make  the  decimal  on  its  right,  thus,  0. 

We  now  consider,  that  625  will  not  go  in  the  10's, 
nor  in  the  100's,  nor  in  the  3,  nor  in  the  37,  nor  in  the 
375,  but  it  will  go  in  the  3753. 

We  must  make  a  trial  at  every  step,  that  is,  every 
time  we  take  in  view  another  place ;  and  we  must  take 
but  one  at  a  time.  In  this  case,  then,  we  shall  have 
0.000006,  the  quotient. 

Divide  3  by  30.  30  will  not  go  in  3  ;  we,  therefore, 
write  0  for  place  of  whole  numbers,  and  then  say  30 
in  30  tenths,  1  tenth  times ;  or,  0.1. 

Divide  .55  by  11. 

11)0.55(0.05 

11  in  0,  no  times ;  11  in  5  tenths,  0  times  ;  11  in  55 
hundredths,  5  hundredths  times. 

It  will  be  observed,  that  we  make  the  decimal  point  in 
the  quotient  as  soon  as  we  ascertain  it ;  not  wait,  and 
then  find  where  it  should  be  by  counting,  &c. — a  rule 
that  we  regard  as  unworthy  of  being  followed  by  all 
those  who  can  use  their  reason. 


EXAMPLES. 

1  Divide  9  by  450.  Ans.  0.02. 

V  Divide  2.39015  by  .007.  Anss.  341.45. 

3  Divide  100  by  .25.  Ans.  400.00. 

4  If  350  pounds  of  beef  cost  $12.25,  what  is  the  cost 
af  one  pound  ?  Ans.  .035. 

5  At  $5.75  per  yard,  how  much  cloth  can  be  pur- 
chased with  $19.50625  ?  Ans.  3.375  yards. 

i  2 


202  APPENDIX. 

6  At  .07  per  cent,  per  annum,  how  much  capital  must 
be  invested  to  yield  $602  ?  Ans.  $8600. 

7  A  benevolent  individual  gave  away  $600  per  annum 
to   charitable   objects,  which  was   .12   of  his   income. 
What  was  his  income  ? 


SECTION  III. 
Multiplication  and  Division  Combined. 

WHEN  it  becomes  necessary  to  multiply  two  or  more 
numbers  together,  and  divide  by  a  third,  or  by  a  product 
of  a  third  and  fourth,  it  must  be  literally  done,  if  the 
numbers  are  prime. 

For  example :  Multiply  19  by  13,  and  divide  that  pro- 
duct by  7. 

This  must  be  done  at  full  length,  because  the  numbers 
are  prime  ;  and  in  all  such  cases  there  will  result  a  frac- 
tion. 

But  when  two  or  more  of  the  numbers  are  composite 
numbers,  the  work  can  always  be  contracted. 

Example :  Multiply  375  by  7,  and  divide  that  pro- 
duct by  21.  To  obtain  the  answer,  it  is  sufficient  to  di- 
vide 376  by  3,  which  gives  125. 

The  7  divides  the  21,  and  the  factor  3  remains  for  a 
divisor. 

Here  it  becomes  necessary  to  lay  down  apian  of  oper- 
ation. 

Draw  a  perpendicular  line,  and  place  all  numbers  that 
are  to  be  multiplied  together  under  each  other,  on  the 
right  hand  side,  and  all  numbers  that  are  divisors  under 
each  other,  on  the  left  hand  side. 

EXAMPLES. 

1  Multiply  140  by  36,  and  divide  that  product  by 
84.  We  place  the  numbers  thus  : 

84      14° 
84       36 


APPENDIX.  203 

We  may  cast  out  equal  factors  from  each  side  of  the 
line  without  affecting  the  result.  In  this  case,  12  will 
divide  84  and  36.  Then  the  numbers  will  stand  thus  : 

140 
3 

But  7  divides  140,  and  gives  20,  which,  multiplied  by 
3,  gives  60  for  the  result. 

2  Multiply  4783  by  39,  and  divide  that  product 
by  13: 

4783 


A* 
** 


3. 


Three    times    4783  must  be  the  result. 

3.  Multiply  80  by  9,  that  product  by  21,  and  divide 
the  whole  by' the  product  of  60X6X  14. 


9 

to  3 

In  the  above,  divide  60  and  80  by  20,  and  14  and  21 
by  7,  and  those  numbers  will  stand  cancelled  as  above, 
with  3  and  4,  2  and  3  at  their  sides. 

Now  the  product  3X6X2,  on  the  divisor  side,  is 
equal  to  4  times  9  on  the  other,  and  the  remaining  3  is 
the  result. 

Hoping  the  reader  now  understands  our  forms,  and 
comprehends  the  true  philosophical  principle,  we  will 
give  no  more  abstract  examples  ;  but  we  will  give  many 
practical  examples,  such  as  might  occur  in  business,  and 
we  prefer  taking  them  from  books,  that  it  may  not  be 
said  we  made  them  expressly  for  this  occasion. 

Again,  it  may  be  observed,  that  this  method  of  opera- 
tion may  serve  for  only  a  few  problems.  We  answer,  it 
will  serve  for  71  out  of  100,  according  to  the  theory  of 
numbers,  as  we  have  seen  there  are  71  composite  num- 
bers in  the  first  hundred,,  and  more  as  they  rise  higher. 
But  the  prime  numbers,  2,  3,  and  5,  are  so  small  and 
manageable  ami  are  factors  in  FO  many  other  numbers, 


204  APPENDIX. 

that  they  may  be  considered  in  as  favorable  a  light  as  the 
composite  numbers — and  for  this  reason  we  may  say, 
75  out  of  every  hundred  problems  can  be  abridged. 

But,  in  actual  business,  the  problems  are  almost  all 
reduceable  by  short  operations ;  as  the  prices  of  articles, 
or  amount  called  for,  always  corresponds  with  some 
aliquot  part  of  our  scale  of  computation. 

This  method  may  not  work  a  great  many  problems  as 
they  are  found  in  some  books,  but  it  will  work  90  out 
of  every  100  that  ought  to  be  found  in  books. 
*In  a  book,  we  might  find  a  problem  like  *his:  - 

What  is  the  cost  of  21b.  7oz.  13dwt.  of  tea  at  7s.  5d. 
per  pound  ?  But  the  person  who  should  go  to  a  store 
and  call  for  31b.  7oz.  and  13dwt.  of  tea,  would  be  a  fit 
subject  for  a  mad-house.  The  above  problem  requires 
downright  drudgery,  which  every  one  ought  to  be  able 
to  perform,  but  such  drudgery  never  occurs  in  business. 

The  following  examples  are  extracted  from  books  in 
common  use,  and  we  mark  them  in  order  that  any  one 
may  find  the  original.  For  instance,  T.  42,  means  Tal- 
bott's  Arithmetic,  page  42  ;  R.  93,  means  Ray's  Arith- 
metic, page  93  ;  E.  123,  means  Emerson's  Arithmetic, 
page  123,  &c. 

EXAMPLES. 

4  How  many  bushels  of  apples  can  be  bought  for  $3, 
at  15  cts.  a  peck?  (R.  92.)  Ans.  5. 

$  Explanation — 3  in  15,  5  times  ;  4 

5     times  5  are  20,  and  20  in  100,  5  times. 


5  A 'farmer  has  91  bushels  of  wheat,  and  he  wishes  to 
put  it  into  bags,  each  of  which  holds  3  bushels  2  pecks ; 
how  many  bags  will  it  take  ?  (R.  92.)  Ans.  26. 


3bu.  2  pe.  =  14pe. 


13 


6  What  is  the  value  of  a  piece  of  gold  weighing  lib. 
3dwt.,  at  12|cts.  per  grain  ?  (R.  92.)          Ans.  $729. 

.1243 
?  \&L  3 


APPENDIX. 


205 


7  At  Sets,  a  pound  what  will  6cwt.  Iqr.  cost?     (R. 
83.)  Ans.  $21. 


8  At  $2.25  a  qr.  what  will  1   ton   Icwt.  cost?  (R. 
93.)  Ans.  $189. 

21  cwt. 

4 

9 

9  At  5cts.  per  oz.,  what  will  7  Ibs.  8oz.  cost?  (R. 
93.)  Ans.  $6. 

10  In  this  example,  we  must  reduce  7lbs.  8oz.  to 
ounces:  7X16+8  is  the  same  as  14X8+8,  or  15X8. 


100 


15 

8  ^=100 
5 


30 

2 

10 


1  1  A  grocer  bought  a  lot  of  cheese,  each  weighing  91bs. 
15oz.,  the  weight  of  the  whole  amounted  to  39cwt.  3qr.; 
how  many  cheeses  were  there  ?  (R.  93.)  Ans.  448. 

qr. 


28 
16 

12  How  many  casks,  each  holding  841bs.,  can  be  fill- 
ed out  of  a  hogshead  of  sugar  weighing  15cwt.  3qr?  (R. 
93.)  Ans.  17. 

13  A  bell  of  Moscow  weighs  288000lbs.;  how  many 
tons?  (R.  93  —  part  of  Ex.  23.) 

Ans.  128  tons.  11  cwt.  Iqr.  20lbs. 

14  "VVe   prefer   dividing   (mentally)   the  pounds  into 
their  obvious  factors. 

7)900 


.  ,       1000  1281 

15  How  many  times  will  a  wheel,  which  is  9ft.  2in. 
18 


206 


APPENDIX. 


in  circumference,  turn  round  in  going  65  miles  ?  (R.  94, 
Ex.  32.)  Ans.  37440. 

65 


X10 


320 
12 


16  What  will  2  square  yards  2  square  feet  of  ground 
come  to  at  Sets,  a  square  inch  ?  (R.  94.)     Ans.  $144 


2  square  yards  2  square  feet=20 
square  feet. 


12 
12 


17  What  will  one  square  yard  of  gilding  cost  at  12.5 
cents  a  square  inch?  (R.  94.)  Ans.  $162. 


ft 
8 


9 
144 


18  What  will  5  yards  2qrs.  of  cloth  cost  at  12|cts.  a 
nail?  (R.  96.)  Ans.  $11. 

8  I  22  qrs. 

19  How  many  coat  patterns,  each  containing  3  yards 
2qrs.,  can  be  cut  out  of  a  piece  of  cloth  containing  70 
Ells  Flemish?  (R.  95.)  Ans.  15. 

H73° 

20  What  will  one  hhd.  of  wine  cost  at  6lcts.  a  gill? 
(R.  95.)  Ans.  $126. 

Observe,  that  6|  =  2TS  ;  and,  as  4  is  a  divisor  to  25,  it 
must  be  put  on  the  opposite  side  of  the  line. 


100 
4 


63 
4 
2 
4 

25 


or, 


16 


63 
4 

2 
4 


21  If    a   person  write    10   minutes   each   day,   how 


APPENDIX. 


207 


much  time  will  that  amount  to  in  4  years  ?   (R.  96.) 

Ans.  10  days  85  hours. 


60 
24 


365 
4 
10 


22  How  many  yards  of  carpeting,  2  feet  6  inches  in 
breadth,  will  cover  a  floor  27  feet  long  and  20  feet  wide  ? 
(T.  98.)  Ans.  72. 


2£=f.     2  is  to  divide  the  5  ;  it  must,         5 
therefore,  go  over  the  line.  3 


20 

27 

2 


23  What  quantity  of  shalloon,  3  quarters  wide,  will 
line  71  yards  of  cloth  that  is  1£  yards  wide?  (T.  98.) 

Ans.  15. 


or, 


15 


N.  B.  Mixed  numbers  are  reduced  to  improper  frac- 
tions, and  the  denominators  thrown  over  the  line. 

24  How  much  land,  at  $2.50  per  acre,  must  be  given 
in  exchange  for  360  acres,  worth  $3.75.  (T.  99.) 

Ans.  $540. 


360 
31 


or, 


90 
3 


25.  What  will  a  bnshel  of  clover-seed  come  to  at  12; 
cts.  a  pint?     (Wilson,  41.)  Ans.  $8. 


12£  cts.=|  of  a  dollar.  The  8  on 
one  side  cancels  8  on  the  other,  and 
leaves  4X2  for  the  answer. 


4  pecks. 


26  Suppose  a  hogshead  of  molasses,  which  cost  $23, 
be  retailed  at  125Cts.  a  quart;  what  is  the  profit  on  it? 
(W.  41.)  Ans.  $8.50. 


208 


APPENDIX. 


63  gallons. 


Sale 
Cost 


31.50 
23 


Profit    8.50 


27  What  will  5  barrels  of  flour  cost  at  3|cts.  per 
pound?  Ans.  $34.30. 

28  How  many  times  is  §  of  a  pint  contained  in  i  of  a 
gallon  ?  (W.  65.) 


Ans.  6|. 


or, 


f 


20 


We  have  already  remarked,  that  denominators  of 
fractions  must  go  over  the  line  from  the  term  to  which 
they  belong. 

29  How  many  times  can  a  vessel,  holding  T9^  of  a 
quart,  be  filled  from  ~  of  a  barrel  containing  31  £  gal- 
lons ?  (W.  66.)  Ans.  46f . 


81* 

5   =•= 


10 


30  If  one  acre  and  20  rods  of  ground  produce  45 
bushels  of  wheat ;  at  that  rate,  how  much  will  nine  acres 
produce  ?  (W.  90.)  Ans.  360. 


la.  20r.  =  180. 


45X8=360. 


45 


N.  B.  We  shall  plan  no  more  problems  in  this  sec- 
tion ;  but  the  following  require  no  real  labor,  save  cor- 
rect reasoning.  When  the  numbers  are  properly  arrang- 
ed, a  few  clips  with  the  pencil,  and  perhaps  a  trifling 
multiplication,  will  suffice. 

31  At  l|cts.  a  gill,  how  many  gallons  of  cider  can  be 
bought  for  $12  ?  (R.  95.)  .       Ans.  25. 

32  How  many  casks,  each  containing  12  gallons,  can 
be  filled  out  of  a  ton  of  wine  ?  (R.  95.)  Ans.  21. 


APPENDIX,  209 

33  A  man  retailed  9  barrels  of  ale,  and  received  for 
it  $129.60  ;  at  what  price  did  he  sell  it  a  pint  ?  (R.  96.) 

Ans,  Sets. 

34  How  much  butter,  at  9cts,  per  pound,  will  pay  for 
12  yards  of  cloth,  at  $2.19  per  yard  ?  (W.  79.) 

Ans.  292 

35  At  45|  dollars  per  acre,  what  will  32  rods  of  land 
come  to?  (W.  79.)  Ans,  $9.10. 

36  How  long  must  a  laborer  work,  at  62;|cts.  a  day, 
to  earn  $25  ?  (W.  78.)  Ans.  40  days. 

37  A  merchant  sold  275  pounds  of  iron,  at  5|cts,  a 
pound,  and  took  his  pay  in  oats,  at  50cts.  a  bushel;  how 
many  bushels  did  he  receive  ?    (Adams,  53.) 

Ans.  38§. 

38  How  many  yards  of  ctoth,  at  $4.66  a  yard,  must 
be  given  for  18  barrels  of  flour,  at  $9.32  a  barrel?  (A. 
53.)  Ans.  36. 

39  How  long  will  it  require  one  of  the  heavenly  bo- 
dies to  move  through  a  quadrant,  at  the  rate  of  43'   12" 
per  minute  ?  (R.  97.)  Ans.  2~  hours. 

40  If  a  comet  move  through  an  arc   of  7°  12-  per 
day,  how  long  would  it  be  in  passing  an  arc  of  180°  ? 
(R.  97.)  Ans.  25  days. 

41.  What  is  the  cost  of  8hhds.  of  wine,  at  5cts.  per 
pint?  Ans,  $201.60. 

42  There  are  30|  square  yards  in  one  perch  of  land  ; 
how  many  perches  are  there  in  363  square  yards  ? 

Ans.  12. 

43  What  will  18 1  yards  cost,  at  75cts.  per  yard  ? 

Ans.  $14.06|. 

44  If  16  persons  receive  $516  for  43  days'  work,  how 
much  does  each  man  earn  per  day  ?  Ans.  75cts. 

45  How  many  times  will  a  wheel,  which  is  12  feet  in 
circumference,  turn  round  in  going  a  mile  ?     Ans.  440. 

46  An  auctioneer  sold  45  bags  of  cotton,  each  contain- 


210  APPENDIX. 

ng  400  pounds,  at  1  mill  a  pound  ;  what  did  the  whole 
come  to?  (R.  61.)  Ans.  $18. 


47  A  mechanic  receives  $90  for  40  daysT  work  —  he 
ivorked  12  hours  each  day;  how  much  was  it  per  hour? 
(R.  73.)  Ans,  ISfcts. 

48.  A  laborer  worked  26  days  for  a  farmer,  at  87|cts. 
per  day,  and  took  his  pay  in  wheat,  at  65cts.  per  bushel  ; 
how  many  bushels  did  he  receive  ?  Ans.  35. 


SECTION  IV. 

IT  is  an  axiom  in  philosophy,  that  equal  causes  pro- 
duce equal  effects ;  and  that  effects  are  always  propor- 
tionate to  their  causes. 

Now,  causes  and  effects  that  admit  of  computation, 
that  is,  involve  the  idea  of  quantity,  may  be  represented 
by  numbers,  which  will  have  the  same  relation  to  each 
other  as  the  things  they  represent. 

Keeping  these  premises  in  view,  then,  we  have  a  uni- 
versal rule,  applicable  to  all  cases  which  can  arise  under 
Proportion,  simple  or  compound,  direct  or  inverse 
namely : 

RULE.— As  any  given  cause  is  to  its  effect,  so  is  any 
required  cause  (of  the  same  kind]  to  its  effect ;  or,  so  is 
another  given  cause,  of  the  same  kind,  to  its  requirea 
effect. 

The  only  difficulty  the  pupil  can  experience  in  thi 
system  of  proportion,  is  readily  to  determine  what  is 
cause,  and  what  is  effect.  But  this  difficulty  is  soon  over 
eome,  when  we  consider,  that  all  action,  of  whatsoevei 
nature,  must  be  cause— and  whatever  is  accomplished  by 
3  that  action,  or  follows  such  action,  must  be  effect. 

y 

EXAMPLES. 

>  If  10  horses,  in  50  days,  consume  128  bushels  oi 


APPENDIX.  211 

oats,  how  many  bushels  will  5  horses  consume  in  90 
days?  (W.  113.)  Ans.  72. 

Here  it  is  evident,  that  the  consumption  of  oats  spoken 
of,  in  both  the  supposition  and  demand,  are  the  true  ef- 
fects ;  and  the  action  of  the  horses,  multiplied  by  the 
days,  must  express  the  amount  of  cause.  We  shall 
therefore  state  it  thus  : 

Cause.         Effect.  Cause.         Effect. 

16       :       128       :  :         5        :         [  ] 
50  90 

We  write  the  factors,  one  under  another,  as  above ; 
their  multiplication  is  understood,  but  rarely  or  never 
actually  accomplished.  The  second  effect  is  an  un- 
known term,  or  answer,  required — a  bracket,  or  blank, 
or  point,  is  left  to  represent  it.  When  found,  the  four 
terms  above  would  be  a  perfect  Geometrical  proportion, 
and  the  product  of  the  extremes  equal  to  the  product  of 
the  means.  In  this  example,  the  product  of  the  means 
is  perfect ;  which  product,  divided  by  the  factors  in  the 
extremes,  will  give  what  is  wanting  in  the  extremes, 
nainly — the  answer. 

Therefore,  agreeably  to  one  mode  of  performing  mul- 
tiplication and  division,  we  draw  a  line  thus,  and  cancel 
down : 

ifi     128 
16        - 

If  $480,  in  30  months,  produce  $84  interest,  what  cap- 
ital, in  15  months,  will  produce  $21  ? 

Now  capital  will  not  produce  interest  without  time ; 
and,  whatever  be  the  rate  per  cent.,  the  same  capital  in  a 
double  time,  will  produce  a  double  interest.  Therefore, 

Cause.        Effect.  Cause.        Effect. 

480        :       84          :  :          [  ]       :        21 
30  15 


212  APPENDIX. 

Here  one  element  of  the  second  cause  is  wanting ;  that 
is,  the  answer  to  the  question. 

In  this  case,  the  extremes  are  complete;  we  will, 
therefore,  divide  the  product  of^the  extremes  by  the  fac- 
tors in  the  mean,  and  the  quotient  will  give  the  definite 
factor,  or  answer,  namely — $240. 

3  If  7  men,  in  12  days,  dig  a  ditch  60  feet  long,  8 
feet  wide,  and  6  feet  deep,  in  how  many  days  can  21 
men  dig  a  ditch  80  feet  long,  3  feet  wide,  and  8  feet 
deep? 

.  It  is  almost  too  plain  for  comment,  that  7  men,  multi- 
plied by  12  days,  must  be  the  first  cause,  and  the  con- 
tents of  the  ditch  they  dig,  the  effect.  Therefore, 

Cause.        Effect.  Cause.        Effect. 

7        :         60  :  :         21  :        80 

12  8  [  ]  3 

6  8 

Here,  as  in  the  preceding  example,  one  of  the  elements 
of  the  second  cause  is  wanting ;  or,  rather  say  a  factor 
in  the  means  of  a  perfect  proportion,  and  can  be  found 
as  above.  Ans.  2§  days. 

5  If  6  men  build  a  wall  in  12  days,  how  long  will  it 
require  20  men  to  build  it  ?  Ans.  3|  days. 

Questions  of  this  kind  are  usually  classed  under  the 
single  rule  of  three  inverse ;  they  do  in  fact,  however, 
belong  to  compound  proportion :  but,  as  one  of  the  terms 
is  the  same  in  the  supposition  as  in  the  demand,  it  may 
be  omitted.  The  term,  in  the  present  example,  is,  one 
wall.  If  we  make  the  number  different  in  the  two 
branches  of  the  question,  or  connect  any  conditions  with 
it,  such  as  lengths,  breadths,  &c.,  it  at  once  falls  under 
compound  proportion  of  necessity,  and  may  be  stated 
thus  : 

Cause.      Effect.  Cause.       Effect. 

6:1          :  :         20       •         1 
12  [] 

5  If  4  men,  in  2^  days,  mow  6|  acres  of  gra^c  by 
•«-• 


APPENDIX.  213 

working  8|  hours  a  day,  how  many  acres  will  15  men 
mow  in  3 1  days,  by  working  9  hours  a  day? 

Ans.  40  y£  acres. 

Cause.       Effect.  Cause.       Effect. 

2'      '       6|         :  :         ll»    ''      C  ] 
81  9 

Let  the  pupil  observe,  that  when  a  correct  statement 
is  made,  there  will  be  the  same  number  of  elements,  or 
factors,  under  the  same  letters,  as  in  the  above  example 
under  each  Cause.  We  have  men,  days,  and  hours,  to 
be  multiplied  together.  When  there  are  fractions  in  any 
of  the  terms,  their  denominators  are  to  be  placed  over 
the  line  from  where  the  term  belongs ;  mixed  numbers 
being  previously  reduced  to  improper  fractions. 

6  If  12  oz.  of  wool  make  1£  yards  of  cloth,  |  of  a 
yard  wide,  how  many  yards,  1|  wide,  will  16  pounds  of 
wool  make  ?  Ans  22 1  yards. 

With  this  example,  some  might  hesitate  as  to  arrang- 
ing it  under  cause  and  effect,  as  the  actors,  those  wh^> 
made  the  cloth,  whether  many  or  few,  have  nothing  to 
do  with  the  question.  But  we  take  the  phrase  of  the 
example  and  say,  The  wool  makes  the  cloth. 

Cause.       Effect.  Cause.       Effect. 

12       :       U         :  :         16 
I  16 

7  If  the  transportation   of  12icwt.   206   miles    cost 
$25.75,  how  far,  at  the  same  rate,  may  3  tons  and  3qrs. 
be  carried  for  $243  ?  Ans.  402f  miles. 

In  this  example,  it  is  indifferent  which  we  take  for  the 
cause  and  which  for  the  effect.  We  may  say,  that  the 
money,  $25.75,  is  the  cause  of  having  the  weight  car- 
ried ;  or,  we  may  say,  that  carrying  the  weight  is  the 
cause  of  purchasing  the  money.  There  are  many  ques- 
tions where  it  is  indifferent  which  we  take  for  cause,  and 


214  APPENDIX. 

which  for  effect.     The  above  example  may  be  stated 


thus : 

Cause. 
25|      : 

Or  thus  : 
Cause. 
12| 
206 

Or  thus : 
Cause. 

206T 
Or  thus : 
Effect. 
243 


Effect. 
12icwt. 
206  miles. 

Effect. 


Cause. 
60| 

C3 

Effect. 


Cause. 
243 


Effect. 
60|cwt, 


Cause.  Effect 
60|     :      243 

CD 

Effect.  Effect. 

OK3  .  OJ.Q 

-£O.T         .         x£-±o 


Cause. 
60! 


Cause. 

121 
206 


These  changes  show,  conclusively,  that  this  method 
of  statement  is  strictly  scientific  and  philosophical ;  and, 
in  all  these  different  arrangements  of  the  terms,  the  same 
terms  are  multiplied  together. 

The  most  that  can  be  said  for  the  common  modes  of 
statements,  in  the  Double  Rule  of  Three,  is,  that  the 
products,  when  the  terms  are  multiplied  out,  are  pro- 
portional. But  the  first  and  second  terms,  taken  as  a 
whole,  express  no  particular  idea  or  thing;  whereas,  in 
this  mode  of  statement,  the  thing — the  philosophical 
idea — is  the  only  sure  guide.  Nor  is  this  all ;  it  is  very 
extensive  and  easy  in  its  application ;  it  will  cover  every 
case  that  can  arise  under  interest — to  find  time,  rate  per 
cent.  &c.,  and  thus  do  away,  or  suspend,  live  or  six 
special  rules,  which  encumber  every  arithmetic.  We 
give  a  few  examp'es  to  apply  this  rule.  • 

8  What  is  the  interest  of  $240  for  3 £  years  at  6  per 
cent.? 

Cause.       Effect.  Cause.      Effect. 

100       :        6       •    :         240     :] 


APPENDIX.  215 

To  obtain  the  answer  from  this  statement,  we  perceive 
that  we  must  multiply  the  means  together — i.  e.  240X6, 
the  rate,  and  that  by  the  3£  years,  the  time — and  divide 
by  100  ;  and  this  is  the  common  rule. 

9  The  interest  of  a  certain  sum  of  money,  at  6  per 
cent.,  for  15  months,  was  $60 ;  what  was  the  sum? 

Ans.  $800. 

Cause.      Effect.^  Cause.      Effect. 

100       :       6     "   :    :        [  ]      :      60 
12  15 

10  If  $800,  in  15  months,  should  gain  $60,  what 
would  be  the  rate  per  cent.?  Ans.  6. 

Cause.      Effect.  Cause.      Effect. 

800      :       60       :    :        100      :[] 
15  12 

11  Eight  hundred  dollars  was  put  out  at  interest,  at  6 
per  cent.,  and  the  interest  received  was  $60;  how  long 
was  it  out?  Ans.  15  months. 

Cause.      Effect.  Cause.       Effect. 

100      :       6         :    :         800      :      60 
12  [] 

12  If  12  men,  working  9  hours  a  day  for  15|  days, 
were  able  to  execute  f  of  a  job,  how  many  men  may  be 
withdrawn  and  the  residue  be  finished  in  15  days  more, 
f  the  laborers  are  employed  only  7  hours  a  day  ?  (W. 
109.)  Ans.  4  men. 

13  The  amount  of  a  note,  on  interest  for  2  years  and 
6  months,  at  6  per  cent.,  is  $690;  required  the  principal. 
(R.  172.)  Ans.  600. 

14  What  is  the  interest  of  $1248  for  16  days?  (R 
162.)  Ans.  $3.28. 

Cause.      Effect.  Cause.       Effect. 

100       :       6        :    :         1248      :      [  ] 


216  APPENDIX, 

This  can  be  cancelled  down  and  made  very  brief. 

15  What  is  the  interest  of  $1200,  for  15  days,  at  6 
per  cent.?  (R.  162.)  Ans.  $3.00. 

16  How  many  men  will  reap  417.6  acres  in  12  days, 
if  5  men  reap  52.2  acres  in  6  days  ?  (T.  156.) 

Ans.  20. 

17  If  a  cellar  22.5  feet  long,  17.3  feet  wide,  and  10.25 
feet  deep,  be  dug  in  2.5  days,  by  6  men  working  12.3 
hours  a  day,  how  many  days,  of  8.2  hours,  should  9 
men  take  to  dig  another  45  feet  long,  44.6  wide,  and 
12.3  deep  ?  (T.  156.)  Ans.  12. 

18  What  is  the  interest  of  160  dollars  for  36  days,  at  7 
per  cent.?     Stated  by  cause  and  effect.        Ans.  $1.12. 

Cause.      Effect.  Cause.      Effect. 

100       :       7        :    :         160      :      [  ] 
12  1.2 

19  The  interest  of  a  certain  sum,  for  36  days,   is 
$1.12 — the  rate  per  cent,  is  7 ;  what  is  the  sum  ? 

Ans.  $160. 

20  The  interest  of  $160  for  36  days,  is  $1.12  ;  wHt 
was  the  rate  per  cent.?  Ans.  7. 

21  The  interest  of  $160,  at  7  per  cent.,  was  $1.1$  ; 
what  was  the  time  ?  Ans.  36  days. 

22  In  what  time  will  any  sum  double  itself  at  6  per 
cent.?     At  any  per  cent.? 

Ans.  At  6  per  cent.,  in  16J  years;  at  any  per  cent., 
divide  100  by  the  per  cent 

23  If  2k  yards  of  cloth,  1J  wide,  cost  $3.37£,  how 
much  will  36£  yards  cost,  1%  yards  wide? 

Ans.  $52.79. 

24  If  4  men  spend  |  of  £  of  f  of  i-J  of  ^30,  in  T7T 
of  T\  of  §4  of  y    of  9  days,  how  many  dollars,  at  6 
shillings  each,  will  21  men  spend  in  ?  of  yf  of  -f  of  T4j 
of  45  days?  (Burnham,  142.)  Ans.  $630. 

25  I  lend  a  friend  $200  for  6  months ;  how  long  ought 


APPENDIX.  217 

he  to  lend  me  $1000,  to  requite  the  favor — allowing  30 
days  to  a  month  ?  Ans.  36  days. 

26  If  1000  men,  besieged  in  a  town,  with  provisions 
for  5  weeks,  allowing  each  man  16  ounces  per  day,  be 
reinforced  with  500  men  more — and  supposing  that  they 
cannot  be  relieved  until  the  end  of  eight  weeks — how 
many  ounces  a  day  must  each  man  have,  that  the  provis- 
ions may  last  them  that  time  ?  (W.  183.) 

Ans.  6J  ounces 

The  advocates  of  this  system  are  of  opinion,  that 
there  are  far  too  many  rules  in  our  common  arithmetics  ; 
and  to  reduce  them,  and  thereby  simplify  the  science, 
they  recommend  that  all  the  problems,  generally  arrang- 
ed under  Profit  and  Loss,  Equation  of  Payments,  &c., 
should  be  solved  by  proportion,  and  arranged  under 
that  head.  In  this  light,  they  are  more  simple  and  in- 
telligible than  they  can  be  made  by  any  special  rules. 

EXERCISES    FOR    PRACTICE. 

1  If  I  buy  cotton-cloth  at  2s.  per  yard,  and  sell  it  at 
2s.  8d,,  what  do  I  gain  per  cent?  (W.  134.) 

Ans.  33£. 

Statement. — If  24  pence  gain  8  pence,  what  will  100 
pence  gain?     Or,     24  :  8  :  :   100  :  Ans. 
Or,       3  :  1  :  :   100  :  Ans. 

2  A  merchant  bought  broadcloth,  at  $5.50  per  yard, 
and  sold  it  for  $6.60  ;  what  did  he  gain  per  cent?  (W. 
135.)  Ans.  20. 

3  If  I  buy  Irish  linen  at  2s.  3d.  per  yard,  how  must 
I  sell  it  to  gain  25  per  cent?  Ans.  2s.  9d.  3h. 

Statement. — If  100  pence  return  125  pence,  how 
much  must  27  pence  return  ? 

Or,     100  :   125  :  :  27  :  Ans. 
Or,       4     :     5     :  :  27  :  Ans. 

4  If,  by  selling  cloth  at  $6.50  per  yard,  I  lose  20  per 
cent.,  what  was  the  prime  cost  of  it  ?  (W.  137.) 

Ans.  $8.12£. 


218  APPENDIX. 

That  is,  if  80  I  now  receive  originally  cost  me  100, 
what  did  6.50  originally  cost? 

5  By  selling  calico  at  37^  cents  a  yard,  50  per  cent 
was  gained;  what  was  the  first  cost?  (R.  185.) 

Ans.  25cts. 

150  :  100  :  :  37|  :  Ans. 
Or,     3     :     2     :  :  37£  :  Ans. 

6  Sold  wine  at  $1.36  per  gallon  and  lost  15  per  cent.; 
what  per  cent,  would  have  been  gained  had  the  wine 
been  sold  for  $1.856  per  gallon  ?  (R.  187.)     Ans.  16. 

7  Bought  126  gallons  of  wine  for  150  dollars,  and  re- 
tailed it  at  20cts.  per  pint ;  what  was  the  gain  per  cent.? 
(T.  129.)  Ans.  34f. 

8  If  $126.50  are  paid  for  llcwt.  Iqr.  25lbs.  of  su- 
gar, how  must  it  be  sold  a  pound  to  make  30  per  cent, 
profit?  (W.  135.)  Ans.  12|cts. 

9  If  I  buy  124cwts.  of  sugar  for  $140,  at  how  much 
must  I  sell  it  per  pound  to  make  25  per  cent.? 

Ans.  12jcts. 

10  If  a  firkin  of  butter,  containing  561bs.  cost  $7,  at 
how  much  must  it  be  sold  per  pound  to  yield  30  per 
cent,  profit?  Ans.  16|cts. 

11  What  is  the  whole  loss,  and  what  is  the  loss  per 
cent.,  in  laying  out  $70  for  hats,  at  $1.75  each,  and  sell- 
ing them  for  25cts.  a-piece  less  than  cost?   (Burnham, 
173.)  Ans.  Whole  loss  $10;  loss  per  100,  14f. 

12  A  merchant  purchases  180  casks  of  raisins,  at  16 
shillings  per  cask,  and  sells  the  same  at  28  shillings  per 
owl.,  and  gains  25  per  cent.;  what  is  the  weight  of  each 
cask?  (B.  174.)  Ans.  80lb. 

We   multiply    180    by    16,   and    to  180 

add  £   for   25  per  cent.,  we  multiply  16 

by   5    and    divide   by   4.      Then    di-  4    •    5 

vide  by  28,  and  it  gives  cwt.;  multiply         28     1J2 
by  112,  and  we  have  pounds;  then  di-       180 
vide  by  180,  and  we  have  pounds  in  each  cask      That 
is,  arrange  the  numbers  as  above,  and  cancel  down. 


APPENDIX.  219 

For  other  examples,  the  student  is  referred  to  the  body 
of  the  work. 


SECTION  V. 

Square  and  Cube  Roots. 

To  work  the  square  and  cube  roots  with  ease  and  fa- 
cility, the  pupil  must  be  familiar  with  the  following  pro- 
perties of  numbers. 

I.  A  square  number,  multiplied  by  a  square  number, 
the  product  will  be  a  square  number. 

II.  A  square  number,  divided  by  a  square  number,  the 
quotient  is  a  square. 

III.  A  cube  number,  multiplied  by  a  cube,  the  product 
is  a  cube. 

IV.  A  cube  number,  divided  by  a  cube,  the  quotient 
will  be  a  cube. 

If  the  square  root  of  a  number  is  a  composite  number, 
the  square  itself  may  be  divided  into  integer  square  fac- 
tors ;  but  if  the  root  is  a  prime  number,  the  square  can- 
not be  separated  into  square  factors  without  fractions. 

N.  B.  Substitute  the  word  cube,  for  square,  in  the 
preceding  sentence,  and  tiie  same  remarks  apply  to  cube 

numbers. 

x 

No  person  can  extract  roots  with  any  tolerable  degree 
of  skill,  without  being  able  to  recognize  the  squares  and 
cubes  of  the  nine  digets  as  soon  as  seen. 


Numbers,       1 

2 

3 

4 

5 

6 

N 

8 

9 

i] 

Squares,          1 

A 

9 

| 

35 

36 

H 

64 

31 

,100 

Cubes,        j    1 

8 

27 

64 

125 

216 

J343| 

512 

729 

luool 

We  here  wish  to  remind  the  reader,  that  the  pupil  is 
supposed  to  understand  the  extraction  of  the   roots   in 


220  APPENDIX. 

the  common  way,  and  we  request  them  not  to  forget 
that  this  is  merely  an  appendix. 


EXERCISES   FOR   PRACTICE. 

1  What  is  the  square  root  of  625  ?  (R.  220.) 

Ans   25. 

If  the  root  is  an  integer  number,  we  may  knpw,  by 
the  inspection  of  the  above  table,  that  it  must  be  25,  as 
the  greatest  square  in  6  is  2,  and  5  is  the  only  figure 
whose  square  is  5  in  its  unit  place. 

Again,  take 625 

Multiply  by 4     4  being  a  square. 

2500 

The  square  root  of  this  product  is  obviously  50 ;  but 
this  must  be  divided  by  2,  the  square  root  of  4,  which 
gives  25,  the  root. 

2  What  is  the  square  root  of  6561  ?  (R.  220.) 

Ans.  81. 

As  the  unit  figure,  in  this  example,  is  1,  and  in  the 
line  of  squares  in  the  above  table,  we  find  1  only  at  1 
and  81,  we  will,  therefore,  divide  6561  by  81,  and  we 
find  the  quotient  81  ;  81  IS,  therefore,  the  square  root. 

3  What  is  the  square  root  of  106729?  (T.  170.) 

Ans.  327. 

As  the  unit  figure,  in  this  example,  is  9,  if  the  number 
is  a  square,  it  must  divide  by  either  9,  or  49.  After  di- 
viding by  9  we  have  11881  for  the  other  factor,  a  prime 
number,  therefore  its  root  is  a  prime  number=  109.  1 09, 
multiplied  by  3,  the  root  of  9,  gives  327  for  the  answer. 

4  What  is  the  root  of  451584  ?  (T.  179.) 

Ans.  672. 

As  the  unit  figure  is  4,  and  in  the  line  of  squares  we 
find  4  only  at  4  and  64,  the  above  number,  if  a  square, 
must  divide  by  4,  or  64,  or  by  both. 


APPENDIX.  221 

We  will  divide  it  by  4,  and  we  have  the  factors  4  and 
112896.  This  last  factor  closes  in  6;  therefore,  by 
looking  at  the  table,  we  see  it  must  divide  by  16,  or  36, 
<fec.  &c. 

We  divide  by  36,  and  we  have  the  factors  36  and 
3136  ;  divide  this  last  by  16,  and  we  have  16  and  196  ; 
divide  this  last  fraction  by  4,  and  we  have  4  and  49. 

Take  now  our  divisors,  and  last  factor,  49,  and  we 
have  for  the  original  number  the  product  of  4X36X  16 
X4X49;  the  roots  of  which  are  2X  6X4X2X7,  the 
products  of  which  are  672,  the  answer. 

5  Extract  the  square  root  of  2025.  (E.  163.) 

Ans.  45. 

Divide  by  25,  and  we  have  its  square  factors,  25  and 
81.  Roots  of  these  factors  are  5X9=45,  the  answer; 

Again,  multiply  by  the  square  number  4,  when  a  num- 
ber ends  in  25,  and  we  have  8100,  root  90,  half  of  which, 
because  we  multiplied  by  4,  the  square  of  2,  is  45,  the 
answer. 

6  What  is  the  square  root  of  390625  ?  (R.  220.) 

Ans.  625. 
390625 
Multiply  by  4, ...  4 

1562500 
Multiply  by  4  again,  4 

6250000 

As  the  number,  independent  of  the  ciphers,  still  ends 
in  25,  we  multiply  again  by  4,  and  we  have  25000000. 
The  root  of  this  is,  obviously,  5000.  Divide  by  2  three 
times,  or  by  8,  and  we  have  625,  the  answer. 

So  far,  some  may  think  this  more  curious  than  useful. 
However  this  may  be,  there  are  problems  where  much 
labor  may  be  saved  by  attending  to  the  foregoing  princi- 
ples. The  following  are  some  of  them : 

Find  a  mean  proportional  between  4  and  256. 

Ans.  32. 


222  APPENDIX. 

Find  a  mean  proportional  between  4  and  196. 

Ans.  28. 
Find  a  mean  proportional  between  25  and  81. 

Ans.  45. 

As  the  above  are  square  numbers,  multiply  their 
square  roots  together  for  the  answers. 

EXAMPLES. 

1  If  484  trees  be  planted  at  an  equal  distance  from 
each  other,  so  as  to  form  a  square,  how  many  will  be  in 
a  row  each  way.  (T.  171.)  Ans.  22. 

Factors  4  and  121     2X11  roots=22. 

2  A  section  of  land,  in  the  Western  states,  is  a  square, 
consisting  of  640  acres  ;  what  is  the  length  in  rods  of 
one  of  its  sides?  (W.  147.)  Ans.  320. 

Nine  out  of  ten  of  our  teachers  would  actually  reduce 
the  acres  to  square  rods,  by  multiplying  by  160,  and  ex- 
tract the  square  root  of  the  product — but  this  would 
show  too  little  attention  to  numbers.  Remove  one  of 
the  ciphers  from  one  number  to  the  other,  and  we  have 
64  to  be  multiplied  by  1600,  both  square  numbers,  whose 
roots  are  8  and  40 — product  320,  the  answer. 

3  What  must  be  the  side  of  a  square  field,  that  shall 
contain   an   area  equal  to  another  field  of  rectangular 
shape,  the  two  adjacent  sides  of  which  are  18  by  72 
rods.  (W.  147.)  •  Ans.  36  rods. 

18  by  72  is  the  same  as  the  half  of  18  by  the  double 
of  72,  or  9  by  144,  square  numbers,  roots  3X12=36, 
the  answer. 

4  A  has  two  fields,  one  containing  10  acres  and  the 
other  12| ;  what  will  be  the  length  of  the  side  of  a  field 
containing  as  many  acres  as  both  of  them  ?  (R.  220.) 

Ans.  60  rods. 

22  ,5  XI 60  is  the  same  as  225X16;  roots  15X4=60, 
the  answer. 

5  What  is  the  mean  proportional  between  24  and  96  ? 

Ans.  48. 


APPENDIX.  223 

6  What  is  the  mean  proportional  between  18  and  32? 

Ans.  24. 

Problems  on  the  Right-angled  Triangle. 

1  The  top  of  a  castle  is  45  yards  high,  and  is  sur- 
rounded with  a  ditch  60  yards  wide ;  required  the  length 
of  a  ladder  that  will  reach  from  the  outside  of  the  ditch 
to  the  top  of  the  castle.  Ans.  75  yards. 

This  is  almost  invariably  done  by  squaring  45  and  60, 
adding  them  together,  and  extracting  the  square  root ;  but 
so  much  labor  is  never  necessary  when  the  numbers 
have  a  common  divisor,  or  when  the  side  sought  is  ex- 
pressed by  a  composite  number. 

Take  45  and  60  ;  both  may  be  divided  by  15,  and 
they  will  be  reduced  to  3  and  4.  Square  these,  9+16 
=25.  The  square  root  of  25  is  5,  which,  multiplied  by 
15,  gives  75,  the  answer. 

2  Two  brothers  left  their  father's  house,  and  went, 
one  64  miles  clue  west,  the  other  48  miles  due  north,  and 
purchased  farms  ;   how  far  are  they  from  each  other? 
(E.  171.)  Ans.  80  miles. 

Divide  by.  -  •   16)64  48 

4     3     16+9=25,5X16=80. 

3  The  hypothenuse  of  a  right-angled  triangle  is  520 
feet,  the  base  312  feet;  what  is  the  perpendicular? 

Ans.  416. 
Divide  by  ....  52)520  312 

2)10       6 

5       3     25— 9=16,  root  4. 
Multiply  by 104 

Answer  .  .  . .^  .  .   416 

4  Required  the  height  of  a  May-pole,  whose  top  be- 
ing broken  off,  struck  the  ground  at  the  distance  of  15 
feet  from  the  foot,  and  measured  39  feet. 

Ans.  75  feet 


224  APPENDIX. 

5  A  hawk,  perched  on   the   top  of  a  perpendicular 
tree,  77  feet  high,  was  brought  down  by  a  sportsman, 
standing  off  14  rods,  on  a  level  with  its  base  ;  what  dis- 

ance,  in  yards,  did  he  shoot?  (W.  149.) 

Ans.  81.154-yards 

If  this  problem  is  worked  with  skill,  it  will  be  requi- 
site to  extract  the  root  of  10  only. 

6  If  the  diagonal  of  a  rectangular  field  is  40  rods,  and 
one  of  the  sides  32,  what  is  the  other  ?  (W.  150.) 

Ans.  24. 
Cubes  and  Cube  Root. 

Cubes,  whose  roots  are  composite  numbers,  may  be 
divided  by  cube  factors.  Cube  numbers,  whose  unit 
figure  is  5,  may  be  multiplied  by  the  cube  number  8, 
and  that  period  reduced  to  ciphers. 

1  What  is  the  cube  root  of  91125  ?  Ans.  45. 
Multiply  by 8 

729000 

Now  729  being  the  cube  of  9,  the  root  of  729000  is 
90 ;  divide  this  by  2,  the  cube  root  of  8,  and  we  have 
45,  the  answer. 

2  The  contents  of  a  cubical  cellar  are  1953.125  cubic 
feet ;  what  is  the  length  of  one  of  its  sides  ?  (R.  225.) 

Ans.  12.5  feet. 
1953.125 
Multiply  by  8,  -  - ..  8 


15625.000 

Multiply  by  8  again,  8 

125.000 

The  cube  root  of  this  is  50  ;  divide  by  4,  because  we 
multiplied  by  8  twice,  and  we  have  12.5  the  answer. 

3  The  number  195112  is  a  cube;  what  is  its  root? 

Ans.  58. 


APPENDIX.  225 

The  cube  numbers  are 

8,     27,     64,     125,     216,     343,     512,     729. 

Comparing  these  numbers  with  195112,  and  we  observe, 
that  the  root,  in  the  place  of  tens,  cannot  be  more  than  5, 
and  the  root,  in  the  place  of  units,  must  be  some  num- 
ber which,  when  cubed,  give  2  for  its  unit  figure — and  8 
js  the  only  figure  possible ;  the  root  of  the  whole  is, 
therefore,  58. 

4  The  number  912673  is  a  cube ;  what  is  its  root  ? 

Ans.  97. 

Observe,  the  root  of  the  superior  period  must  be  9, 
and  the  root  of  the  unit  period  must  be  some  number 
which  will  give  3  for  its  unit  figure  when  cubed,  and  7 
is  the  only  figure  that  will  answer. 

In  this  manner,  we  can  speedily  and  easily  obtain  the 
cube  roots  of  all  cube  numbers  containing  not  more  than 
two  periods,  or  determine  whether  they  are  cubes  or 
surds. 

The  following  numbers  are  cubes ;  required  their 
roots. 

1  What  is  the  cube  root  of  59319  ?  Ans.  39. 

2  What  is  the  cube  root  of  79507  ?  Ans.  43. 

3  What  is  the  cube  root  of  117649?  Ans.  49. 

4  What  is  the  cube  root  of  110592  ?  Ans.  48. 
5.  What  is  the  cube  root  of  357911  ?           Ans.  71. 
6  What  is  the  cube  root  of  389017  ?           Ans.  73. 
8  What  is  the  cube  root  of  571787  ?            Ans.  83. 

When  a  cube  has  more  than  two  periods,  it  can  gener- 
erally  be  reduced  to  two  by  dividing  by  some  one  or  more 
^f  the  cube  numbers,  unless  the  root  is  a  prime  number. 

The  number  4741632,  is  a  cube;  required  its  root. 
He  re  we  observe,  that  the  unit  figure  is  2 ;  the  unit  fig- 
i  re  of  the  root  must,  therefore,  be  the  root  of  512,  as 
that  is  the  only  cube  of  the  9  digits  whose  unit  figure  is 
2.  The  cube  root  of  512  is  8  ;  therefore,  8  is  the  unit 
figure  in  the  root,  and  the  root  is  an  even  number,  and 


226  APPENDIX. 

can  be  divided  by  2 — and,  of  course,  the  cube  itself  can 
be  divided  by  8,  the  cube  of  2. 

8)4741632 

592704 

Now,  as  the  first  number  was  a  cube,  and  being  di- 
vided by  a  cube,  the  number  592704  must  be  a  cube,  and, 
by  inspection,  as  previously  explained,  its  root  must  be 
84,  which,  multiplied  by  2,  gives  168,  the  root  required. 

The  number  13312053,  is  a  cube ;  what  is  its  root  ? 

Ans.  237. 

As  there  are  three  periods,  there  must  be  three  figures, 
units,  tens,  and  hundreds,  in  the  root ;  the  hundreds  must 
be  2,  the  units  must  be  7.  Let  us  then  divide  the  2d 
figure,  or  the  tens,  in  the  usual  way,  and  we  have  237 
for  the  root. 

Again,  divide  13312053  by  27,  and  we  have  493039 
for  another  factor.  The  root  of  this  last  number  must 
be  79,  which,  multiplied  by  3,  the  cube  root  of  27,  gives 
237,  as  before. 

The  number  18609625  is  a  cube ;  what  is  its  root? 

As  this  cube  ends  with  5,  we  will  multiply  it  by  8  : 

18609625 
8 


148877000 

As  the  first  is  a  cube,  this  product  must  be  a  cube ;  and, 
as  far  as  labor  is  concerned,  it  is  the  same  as  reduced  to 
two  periods,  and  the  root,  we  perceive  at  once,  must  be 
530,  which,  divided  by  2,  gives  265  for  the  root  re- 
quired. 

N.  B.  It  a  number,  whose  unit  figure  is  5,  be  mult£ 
plied  by  8,  -md  does  not  result  in  three  ciphers  on  the 
right,  the  number  is  not  a  cube. 

To  find  the  Approximate  Cube  Root  of  Surds. 
The  usual  way  of  direct  extraction,  is  too  tedious  to 


APPENDIX.  227 

be  much  practiced,  if  any  shorter  method  can  possibly 
be  obtained.  By  the  invention  of  logarithms,  a  very 
short  method  has  been  found;  but,  before  that  event, 
several  eminent  mathematicians  bestowed  much  time  and 
labor  to  obtain  short  practical  rules — and  some  of  their 
rules  are  too  ingenious  and  useful  to  be  lost,  notwith- 
standing the  invention  of  logarithms  has  nearly  super- 
ceded  their  absolute  value  in  practice. 

There  is  no  exact  and  constant  relation  between  pow- 
ers and  their  roots ;  for  this  reason,  all  rules  (save  by 
logarithms,  and  the  direct  and  tedious  one,)  must  be  more 
or  less  approximate;  but,  nevertheless,  with  common 
judgment  and  care,  we  can  arrive  at  results  as  near  as  by 
the  direct  methdd. 

In  order  to  obtain  a  rule,  let  us  take  two  cube  numbers, 
as  near  in  value  to  each  other  as  practicable,  and  compare 
them  with  their  roots. 

216000  and  226981  are  cubes  ;  their  roots  are  60  and 
61. 

Now  216000  is  not  to  226981  as  60  to  61.  But  let 
us  double  the  first  and  add  it  to  the  second,  and  double 
the  second  and  add  it  to  the  first,  and  we  shall  have 
658981  and  669962,  which  are  to  each  other  very  nearly 
as  60  to  61. 

Or,  by  the  principles  of  proportion,  the  first  is  to  the 
difference  between  the  first  and  second,  as  is  the  third  to 
the  difference  between  the  third  and  fourth.  That  is, 
658981  :  10981  :  :  60  :  1  very  nearly.  Now  one  is 
the  difference  between  the  two  roots,  and  if  the  last  root, 
or  61,  was  unknown,  this  proportion  would  give  it  very 
nearly. 

EXAMPLES. 
1.  Required  the  cube  root  of  66. 

The  cube  root  of  64  is  4.  Now  it  is  manifest, 
that  the  cube  root  of  66  is  a  little  more  than  4,  and 
by  taking  a  similar  proportion  to  the  preceding,  we 
have 


228  .  APPENDIX. 

64X2  =  128      2X66=132 
66  64 

194        :        196  :  :  4  :  to  root  of  06. 
Or,         194     :     2     :     :     4     :     to  a  correction. 

194)8.0000(0.04124 
176 


240 
194 


460 

388 

720 

Therefore,  the  cube  root  of  66  is  4.04124 

2  Required  the  cube  root  of  123. 

Suppose  it  5 ;  cube  it,  and  we  have  125. 

Now  we  perceive,  that  the  cube  of  5  being  greater 
than  123,  the  correction  for  5  must  be  subtracted 

2X125=250      246 
Add 123       125 



As 373  :  371  :  :  5  :  root  of  123. 

Or,       373     :     2     :  :     5     :     correction  for  5. 

373)10.0000(0.02681 
746 

2  540  From  5.00000 

2238  take     0.02681 


3020  Ans.    4.97319 

2984 

360 


APPENDIX.  229 

From  what  precedes,  we  may  draw  the  following 

RULE.— Take  the  nearest  rational  cube  to  the  given 
number,  and  call  it  the  assumed  cube;  or,  assume  a 
root  to  the  given  number  and  cube  it.  Double  the  as- 
sumed cube  and  add  the  number  to  it ;  also,  double,  the 
number  and  add  the  assumed  cube  to  it.  Take  the 
difference  of  these  sums,  then  say,  J?s  double  of  the  as- 
sumed cube,  added  to  the  number,  is  to  this  difference^ so 
is  the  assumed  root  to  a  correction. 

This  correction,  added  to  or  subtracted  from  the  assum- 
ed root,  as  the  case  may  require,  will  give  the  cube  root 
very  nearly. 

By  repeating  the  operation  with  the  root  last  found  as 
an  assumed  root,  we  may  obtain  results  to  any  degree  of 
exactness ;  one  operation,  however,  is  generally  suf- 
ficient. 

3  What  is  the  cube  root  of  28"?      Ans.  3.03658-|-. 

4  What  is  die  cube  root  of  26?      Ans.  2.96249-}-. 

5  What  is  the  cube  root  of  214  ? 

Ans.  5.98142-h 

6  What  is  the  cube  root  of  346  ? 

Ans.  7.02034-f-. 

The  above  being  very  near  integral  cubes — that  is,  28 
and  26  are  both  near  the  cube  number  27,  214  is  near 
216,  &c.,  all  numbers, very  near  cube, numbers  are  easy 
of  solution. 

We  now  give  other  examples,  more  distant  from  inte- 
gral cubes,  to  show  that  the  labor  must  be  more  lengthy 
and  tedious,  though  the  operation  is  the  same. 

EXAMPLES. 

1   What  is  the  cube  root  of  3214?     Ans.  14.75758. 

Suppose  tho  root  is  15 — its  cube  is  3375,  which,  being 
greater  than  3214,  shows  that  15  is  too  great;  the  cor- 
rection will,  therefore,  be  subtractive. 


230  APPENDIX. 

By  the  rule,  9964  :  161  :  :  15  :  0.2423,  the  cor- 
rection. 

Assumed  root, 15.0000 

Less 2423 

Root  nearly ...«-..  14.7577 

Now  assume  14.7  for  the  root,  and  go  over  the  opera- 
tion again,  and  you  will  have  the  true  root  to  8  or  10 
places  of  decimals. 

2  What  is  the  cube  root  of  14760213677  1 

Ans.  2453. 

Suppose  the  root  2400,  &c.  Take  the  correction  to 
the  nearest  unit,  and  you  will  find  it  53. 

3  What  is  the  cube  root  of  980922617856? 

Ans.  9936. 
Suppose  the  root  to  be  10000. 

4  What  is  the  cube  root  of  9  ?  Ans.  2.08008. 

5  What  is  the  cube  root  of  9^?  Ans.  2.092-f 

6  What  is  the  cube  root  of  41  ?        Ans.  3.44S2-J-. 

When  it  is  requisite  to  multiply  several  numbers  to- 
gether and  extract  the  cube  root,  try  to  change  them  into 
cube  factors,  and  extract  the  root  before  the  multiplica- 
tion. 

EXAMPLES. 

1  What  is  the  side  of  a  cubical  mound  equal  to  one 
288  feet  long,  216  feet  broad,  and  48  feet  high?    (R. 
225.) 

The  common  way  of  doing  this,  is  to  multiply  these 
numbers  together  and  extract  the  root,  a  lengthy  opera- 
tion. But,  observe,  that  216  is  a  cube  number,  and  288 
=2X12X12,  and  48=4X12;  therefore,  the  whole 
product  is  216X8X12X12X12.  Now  the  cube  root 
of  216  is  6,  of  8  is  2,  and  of  123  is  12,  and  the  product 
of  6X2  X  12  =  144,  the  answer. 

2  Required    tlie    cube  root  of  the  product   of   448 
X392,  in  a  brief  manner.  Ans.  56. 

3  If  you  have  a  pile  of  wood  32  feet  long,  4  feet 


APPENDIX.  231 

wide,  and  4  feet  high,  what  v/ould  be  the  side  of  a  cubic 
pile  containing  the  same  quantity  ?  Ans.  8  feet. 

Proposition — The  solid  contents  of  cubes  or  spheres 
are  to  each  other  as  the  cubes  of  their  like  dimensions. 
(See  Geometry.) 

EXAMPLES. 

1  Mercury  is  about  2000  miles  in  diameter,  and  the 
earth  about  8000;  what  is  their  relative  magnitudes  ? 

Ans.  As  1  to  64. 

2  Mars  is  about  4000  miles  in  diameter,  the  earth 
8000  ;  what  is  their  relative  magnitudes  ? 

Ans.  As  1  to  8. 

In  the  preceding  examples  we,  of  course,  do  not  cube 
the  numbers  given,  but  their  smallest  integral  relations. 

3  The  diameter  of  the  earth,  to  that  of  the  sun,  is 
nearly  as  1  to  Ills  ;  what  is  their  relative  magnitudes, 
or  bulks?  Ans.  As  1  to  1384472  nearly. 

4  If  a  ball,  6  inches  in  diameter,  weigh  32lbs.,  what 
will  be  the  weight  of  a  ball,  of  the  same  metal,  whose 
diameter  is  3  inches  ?  (R.  225)  Ans.  41bs. 

6:3::        2:1 
8     :     1     :  :     32     :     4 

5  Suppose  an  iron  ball,  of  4  inches  in  diameter,  to 
weigh  9   pounds ;   required   the  weight  of  a  spherical 
shell  of  9  inches  in  diameter,  and  1  inch  thick. 

Ans.  541bs.  4oz. 

6  If  a  cable,  12  inches  about,  require  an  anchor  of 
18cwt.,  of  what  weight  must  an  anchor  be  for  a  15  inch 
cable?  (Pike,  211.)  Ans  35cwt.  15lbs. 


SECTION  VI. 
Mensuration,  Gauging,  fyc. 

N.  B.  EVERY  problem  that  follows,  can  be  done  with 
much  less  labor  than  they  are  usually  done. 


232  APPENDIX. 

EXERCISES    FOR   PRACTICE 

1  What  is  the  difference  of  area  between  a  square  of 
40  rods  on  a  side,  and  an  equilateral  rhombus  of  40  rods 
to  a  side,  but  of  a  perpendicular  altitude  of  only  34  rods  ? 
(W.  175.)  Ans.  240  rods. 

2  There  is  a  barn,  50  feel  "by  36,  and  20  feet  high  to 
the  eaves ;  how  many  boards  will  it  take  to  cover  the 
body,  if  the  boards  were  all  15  inches  wide  and   10  feet 
long  ?  Ans.  275-j-  boards. 

3  On  a  base  of  120  rods  in  length,  a  surveyor  wished 
to  lay  off  a  rectangular  lot  of  land,  to  contain  60  acres  ; 
what  distance  in  rods  must  he  run  out  from  his  base  line  ? 
(W.  175.)  Ans.  80  rods. 

4  How  many  square  yards  in  a  triangle,  whose  base 
is  48  feet,  and  perpendicular  height  254  feet?  (W.  175.) 

Ans.  67s  yards. 

5  A  man  bought  a  farm  198  rods  long,  and  150  rods 
wide,  and  agreed  to  give  $32  per  acre  ;   what  did   the 
farm  come  ta?  Ans.  $5940. 

N.  B.  Make  no  attempt  to  compute  the  number  of 
acres  definitely. 

6.  If  the  forward  wheels  of  a  coach  are  4  feet,  and 
the  hind  ones  5  feet  in  diameter,  how  many  more  times 
will  the  former  revolve  than  the  latter  in  going  a  mile  ? 
(W.  176.)  Ans  84. 

N.  B.  In  this  problem  use  7  to  22. 

7  How  many  square  feet  in  a  board,  2  feet  wide  at  the 
larger,  1  foot  8  inches  at  the  smaller  end,  and   14  feet 
long  ?  (W.  176.)  Ans.  25|  feet. 

8  The  plate  supporting  the  rafters  of  a  house,  being 
40  feet  long,  14  inches  wide,  and  8  inches  thick,  how 
many  solid  feet  does  it  contain?  (W.  177.) 


Ans.  31|  feet. 


12 
12 
12 


40 
12 
14 
8 


inches 


APPENDIX.  233 

Cancel  down,  and  this  is  the  form  for  all  solids. 

9  If  a  pile  of  wood  be  60  feet  long,   12  feet  high, 
and  6  feet  wide,  how  many  cords  does  it  contain  ? 

Ans  33?  cords. 

10  The  bin  of  a  granary  is  10  feet  long,  5  feet  wide, 
and  4  feet  high  ;  allowing  the  cubical  contents  of  a  dry 
gallon  to  be  268|  cubic  inches,  how  many  bushels  of 
grain  will  it  contain  ?  Ans.  160*. 

1 1  If  you  wanted  a  bin  to  contain  twice  as  much  as 
mentioned  in  the  last  problem,  with  a  length  of  12  feet, 
and  a  breadth  of  6  feet,  of  what  height  must  it  be  ?  (W. 
177.)  Ans.  5|  feet. 

12  A  canal  contractor  engaged  to  excavate  2  miles  of 
canal  across  a  plane,  at  Sots  per  cubic  yard — the  canal  to 
be  54  feet  wide  at  top,  40  at  bottom,  and  4$  feet  deep ; 
what  did  it  amount  to  ?  Ans.  $6617.60. 

13  There  is  a  circular  cistern,  of  uniform  diameter, 
whose  depth  is  8  feet,  and  diameter  5  feet;  what  is  its 
capacity,  allowing  231   inches  to  the  gallon,  and   how 
much  would  its  capacity  be  increased  by  adding  6  inches 
to  its  diameter  ?          .        C  1175.04  gallons  its  capacity, 

S>  I       246|  gallons  increase. 

14  What  would  be  the  produce  of  a  kernel  of  wheat 
in    11    years,    at   20    fold,    the    produce    of  each    year 
being    sowed    the    next-?— allowing    5000   kernels    to   a 
quart?  (W.  166.)  Ans.  64000000  bushels. 

N.  B.  If  we  blindly  perform  all  the  labor  indicated  by 
set  rules,  the  above  would  be  a  tedious  operation  ;  but  it 
is  extremely  brief  in  the  hands  of  a  skillful  operator. 

15  The  length  of  a  room  being  20  feet,  its  breadth  14 
feet  6  inches,  and  its  height  10  feet  4  inches ;  how  much 
will  the  coloring  come  to  at  27cts.  per  square  yard,  de 
ducting  a  fire-place  of  4  feet  by  4  feet  4  inches,  and 
tw.  windows,  each  6  feet  by  3  feet  2  inches?  (R.  232.) 

Ans.  $19.73. 

1 6  What  will  the  paving  of  a  foot-path  come  to,  at  1 8 
cents  per  square  yard,  the  length  being  35  feet  4  inches, 
and  the  breath  8  feet  3  inches  ?  (R.  233.)     Ans.  $5.83. 


234  APPENDIX. 

17  What  will  it  cost  to  roof  a  building  40  feet  long, 
the  rafters  on  each  side  being  18  feet  6  inches  long,  at 
$3.50  per  100  square  feet?  (R.  233.)       Ans.  $51.80. 

18  There  is  a  block  of  marble,  in  the  form  of  a  paral- 
lelepiped, whose  length  is  3  feet  2  inches,  breadth  2  feel 
8  inches,  and  depth  2  feet  6  Inches ;  what  will  it  cost  at 
Slcts.  per  cubic  foot?  (R.  234.)  Ans.  $17.10. 

19  What  will  it  cost  to  build  a  wall  320  feet  long,  6 
feet  high,  and  15  inches  thick,  allowing  20  bricks  to  the 
solid  foot,  at  $5T8^  per  thousand  bricks  ?  (R.  234.) 

Ans.  $282. 

20  How  many  bricks  8  inches  long,  4  inches  wide, 
and  2^  inches  thick,  will  it  take  to  build  a  wall   120  feet 
long,  8  feet  high,  and  1  foot  6  inches  thick  ?  (R.  234.) 

Ans.  34560. 

21  What  will  it  cost  to  build  a  brick  wall  240  feet 
long,  6  feet  high,  and  3  feet  thick,  at  $3.25  per  1000 
bricks — each  brick  being  9  inches  long,  4  inches  wide, 
and  2  inches  thick  ?  (R.  234.)  Ans.  $336.96. 

22  A  ship's  hold  is  75£  feet  long,  18£  wide,  and  7| 
deep;  how  many  bales  of  goods  3£  feet  long,  2|  deep, 
and  2|  wide,  may  be  stowed  therein,  leaving  a  gangway, 
the  whole  length,  of  3?  feet  wide  ?  (Pike,  470.) 

Ans.  385,4-f-. 

N.  B.  Do  this  by  one  operation — after  taking  out  the 
gangway  mentally,  by  subtraction. 

23.  A  stick  of  timber  is  16  inches  broad  and  8  inches 
thick  ;  how  many  feet  in  length  must  be  taken  to  make 
20  solid  feet?  Ans.  22&. 

24  There  is  a  square  pyramid,  each  side  of  whose 
base  is  30  inches,  and  whose  perpendicular  height  is  120 
inches,  to  be  divided  by  sections,  parallel  to  its  base,  into 
three  equal  parts  ;  required  the  perpendicular  height  of 
each  part.  (P.  371.) 

Ans.  The  height  of  the  lower  section  is  15.2  inches  ; 
the  height  of  the  middle  section  is  21.6  inches ;  the  height 
of  the  top  section  is  83.2  inches. 

N.  B.  In  solving  this  problem,  remember  that  solids, 


APPENDIX.  235 

of  the  same  shape,  are  to  each  other  as  the  cubes  of  their 
like  sides. 

25  A  man  wishes  to  make  a  cistern  of  8  feet  in  diameter, 
to  contain  60  barrels,  at  32  gallons  each,  and  231  cubic 
inches  to  a  gallon ;  what  shall  be  the  depth  of  the  cis- 
tern ? 

60X32X231  gives  the  cubic  inches  the  cistern  is  to 
contain. 

This  divided  by  the  circular  end,  expressed  in  in- 
ches, will  give  the  depth  in  inches. 

8X12X22 
.JNow, =  the  circumference  in  inches. 

But  the  circumference  of  any  circle,  multiplied  by  | 
of  its  diameter,  giver  its  area. 

8X12X22X24 
Then, .... =  the  area. 

Hence, $ 


7X7X  5=245  ;  which,  divided  by  4,  gives  6U  inches 
for  the  answer. 


SECTION  VII. 

Miscellaneous  Examples. 

1  One-half,  one-third,  and  one-fourth  of  a  certain  num 
ber,  added  together,  make  130  ;  what  is  the  number? 

Ans.  120. 

To  solve  this  by  arithmetic,  we  must  consider  the 
number  as  the  whole  of  a  thing,  or  a  unit.     Then  5-f 
-r-?  =  T62+T42"i"T32  or  VI-     Then,  by  proportion,  if  \ 
make  130,  what  will  1,  or  4-5,  make  ?     That  is, 


130 


T* 

-'• "V 


236  APPENDIX. 

By  multiplying  the  first  and  last  terms  by  12,  the  pro- 
portion becomes 

13     :     130     :  :     12 
Or, .1     :       10     :  :     12 

2  One-fourth  of  a  certain  number  exceeds  one-sixth 
of  the  same  number  by  20 ;  what  is  the  number  ? 

Here,  again,  the  number  must  be  considered  as  a  sin- 
gle thing ;  then  $  of  it  is  not,  strictly,  one-fourth  of  a 
unit,  but  |  of  that  number,  or  that  thing.  In  algebra, 
this  thing,  or  number,  would  be  represented  by  some  let- 
ter, as  x  or  y — and  the  question  is  strictly  an  algebraical 
one.  But  all  such  questions  in  algebra,  can  be  solved 
by  fractions  and  proportion  in  arithmetic ;  and,  indeed, 
all  questions,  that  involve  simple  equations  only,  can  be 
resolved  by  arithmetic — but  by  algebra  they  are  much 
easier. 

In  days  that  we  re,  we  always  found  a  rule  in  arithme- 
tic called  Position,  which  included  such  problems  as  the 
preceding,  and  some  few  others^  but  could  not  take  in  any 
questions  involving  powers,  or  roots — as.  powers  and 
roots  are  not  in  arithmetical  proportion  to  each  other. 

For  example,  16  and  64  are  square  numbers,  and  their 
square  roots  are  4  and  8,  or  as  1  to  2  ;  but  the  numbers 
themselves,  16  and  64,  are  to  each  other  as  1  to  4,  a  dif- 
ferent relation  from  the  roots. 

Example  :  A  man,  having  a  purse  of  money,  being 
asked  how  much  was  in  it,  answered,  The  square  root 
of  it,  added  to  the  half  of  it,  made  220  dollars ;  how 
much  was  in  the  purse.? 

It  is  evident,  that  this  question  must  be  excluded  from 
any  proportional  operation ;  for,  unless  we  first  suppose 
the  right  number,  the  result  of  the  supposition  will  not 
be  to  the  given  result  as  the  supposed  number  to  the  true 
number — and  when  this  proportion  fails,  supposition, 
that  is,  "Position  fails  ;"  and  if  we  suppose  the  true 
number,  it  is  then  an  operation  of  chance,  and  is,  in  fact, 
no  problem  at  all. 

True  it  is,  we  can  give  a  rule,  or  rather  give  orders 
which,  followed,  will  reduce  the  problem,  but  it  will  not 


APPENDIX.  237 

be  arithmetic;  and,  to  put  into  an  arithmetical  work 
what  is  not  arithmetic,  we  hold  to  be  deceptive  and  im- 
proper. 

We  have  remarked,  that  the  solution  of  these  prob- 
lems are  much  easier  by  algebra  than  by  arithmetic; 
but  we  would  remind  the  pupil,  that  the  solution  of  a 
problem  is  a  small  object  compared  with  a  principle  in- 
volved in  a  solution,  or  with  a  knowledge  of  the  science 
of  numbers.  As  one  step  to  secure  this  latter  object,  we 
give  a  few  more  such  problems. 

3  A  post  is  J  in  the  earth,  |  in  the  water,  and  13  feet 
above  the  water  ;  what  is  the  length  of  the  post  ? 

Ans.  35  feet. 

Add  ^  and  f  ;  not  ^  and  ~  of  the  number  1,  but  i  and 
f  of  the  whole  post.  These  parts,  added  together,  make 
||  ;  the  remaining  ij  must  be  13  feet.  Then,  by  pro- 
portion, 


35 

35 


Or,  .....  13     :     13     :  :     35     :     35  the  answer. 

4  A  and  B  have  the  same  income.    A  contracts  an  an- 
nual debt  amounting  to  ~  of  it,  B  lives  upon  |  of  it  ;  at 
the  end  of  ten  years,  B  lends  to  A  money  enough  to  pay 
off  his  debts,  and  has  160  dollars  to  spare;  what  is  the 
income  of  each  ?  Ans.  $280. 

If  B  lives  on  |,  he  saves  ^  ;  out  of  this  he  pays  A's 
debts,  4.  Hence,  from  jr  subtract  17,  and  there  remains 
?2j.  This,  in  10  years,  is  worth  160  dollars;  therefore, 
in  1  year  it  is  worth  16  dollars.  Now,  by  proportion, 

/T  :  16  :  :  ||  :  the  answer  ; 
Or,  •  •  2  :  16  :  :  35  :  the  answer; 
Or,  .  .  1  :  8  :  :  35  :  280,  the  answer. 

5  Of  the  trees  in  an  orchard,  £  are  apple  trees,  -i 
pear  trees,  and  the  remainder  peach  trees  —  which  are  20 
more  than  |  of  the  whole  ;  what  is  the  whole  number  in 
the  orchard  ?  Ans.  800. 

The  apple,  the  pear,  and  the  peach  trees,  make  the 
whole;  therefore,  add  +  '-=+-=- 


238  APPENDIX. 

This  wants  •/•$,  or  ^L,  of  being  the  whole ;  therefore,  we 
must  conclude,  that  the  20,  not  taken  into  the  account, 
is  ¥V  of  the  whole.  Hence,  20X40=800 ;  or,  by  pro- 
portion, 

TV     :     20     :  :     }«{.     :     answer. 
That  is,    1      :     20     :  :     40     :     800,  the  answer. 

6  A,  B,  and  C,  would  divide  $200  between  them,  so 
that  B  may  have  $6  more  than  A,  and  C  $8  more  than 
B  ;  how  many  dollars  for  each  ?  (R.  229.) 

Ans.  A  60,  B  66,  C  74. 

Observe,  that  A  has  the  least  sum,  B  86  more  than  A, 
and  C  $14  more  than  A ;  hence,  $20  is  to  be  reserved, 
and  the  remaining  180  to  be  divided  equally  among 
them;  which,  of  course,  gives  $60  to  A,  and  $60-}-6  to 
B,  and  60+6+8  to  C. 

7  A  gentleman  bought  a  chaise,  horse,  and  harness  for 
$378 — the  horse  came  to  twice  the  price  of  the  harness, 
and  the  chaise  to  twice  the  price  of  both  the  horse  and 
harness  ;  what  did  he  give  for  each  ?  (R.  229.) 

This  problem  is  generally  given  under  position,  but  it 
is  really  one  in  simple  division.  Take  the  idea  of  shares. 
Divide  the  money  into  shares  :  it  will  take  1  share  to 
purchase  the  harness,  2  shares  to  purchase  the  horse,  6 
shares  to  purchase  the  chaise  ;  therefore,  divide  the  whole 
into  9  shares,  and  we  shall  have  $42  for  one  share,  i.  e. 
$42  for  the  harness,  $84  for  the  horse,  $252  for  the  chaise. 

In  conclusion,  we  would  caution  the  young  arithme- 
tician against  imbibing  the  idea,  that  he  understands  arith- 
metic, from  works  on  arithmetic  alone.  We  must  rise 
above  a  plane,  to  have  a  fair  view  of  the  objects  on  its 
surface,  and  we  must  rise  above  arithmetic  before  we  can 
understand  all  its  scientific  relations. 

Nor  must  we  conclude  that  we  are  perfect  in  num- 
bers, because  we  may  excel  our  comrades  in  disentang- 
ling knotty  questions.  Science  is  a  different  thing  from 
acuteness  at  solving  intricacies ;  and  all  men.  of  true 
science,  more  or  less  despise  all  things  intended  to  puz- 
zle, for  there  are  enough  objects  of  useful  inquiry  and 
investigation,  on  which  to  expend  all  our  powers  of  mind. 


APPENDIX.  239 

Algebra  is  but  a  continuation  of  arithmetic;  and  as 
soon  as  we  acquire  a  good  practical  knowledge  of  arith- 
metic, so  as  to  have  a  clear  comprehension  of  fractions 
and  general  proportion,  we  may,  yea,  we  should,  com- 
mence algebra.  But  when  we  advance  in  algebra,  we 
should  be  careful  not  to  look  down  with  the  spirit  of  con- 
tempt on  arithmetic — we  may  study  the  one  in  order  to 
understand  the  other.  The  following  properties  of  num- 
bers, the  student  must  take  as  facts,  unless  he  is  an  alge- 
braist. They  cannot  be  demonstrated  without  the  aid 
of  that  science.* 

1  If  from  any  number,  the  sum  of  its  digits  be  sub- 
tracted, the  remainder  is  divisible  by  9. 

Take,  for  example,  34 ;  subtract  7,  and  we  have  27, 
which  is  divisible  by  9.  Again,  take  438  ;  subtract  the 
sum  of  4+3+8=15,  and  we  have  423=9X47,  and  so 
with  any  other  number. 

2  If  the  sum  of  the  digits  of  any  number  be  divisi- 
ble by  9,  the  number  itself  is  divisible  by  9. 

Thus  the  numbers  72,  81,  99,  171,  387,  51489,  &c., 
the  sum  of  whose  digits  is  divisible  by  9,  are  themselves 
divisible  by  9. 

All  numbers  divisible  by  9,  are,  of  course,  divisible 
by  3. 

3  If  the  sum  of  the  digits  of  any  number  be  divisi- 
ble by  3,  then  the  number  itself  is  divisible  by  3. 

Thus  18,  27,  54,  75,  111,  123,  258,  &c.  &c.,  are  all 
divisible  by  3. 

4  If  from  any  number  the  sum  of  the  digits  stand- 
ing in  the  ODD  places  be  subtracted,  and  the  sum  of  the 
digits  standing  in  the  EVEN  places  be  added,  then  the 
result  is  divisible  by  11. 

Take  any  number,  say  785432  ;  then  subtract  the  sum 
of  2+4+8  =  14,  and  add  3+5+7=15,  or,  in  this  ex- 
ample, add  1,  and  we  have  785433  =  11X71403. 


*  The  demonstrations  of  these  properties  may  be  found  in  Bridge's 
Algebra,  section  XLII. 


240  APPENDIX. 

5  If  the  sum  of  the  digits  standing  in  the  EVEN  pla- 
ces, be  equal  to  the  sum  of  the  digits  standing  in  the 
ODD  places,  then  the  number  is  divisible  by  11. 

Thus  the  numbers  121,  363,  12133,  48422,  &c.,  are 
all  divisible  by  11. 

Our  numbers  increase  in  a  ten-fold  proportion,  from 
the  right  to  the  left,  which  is  called,  the  root  of  the 
scale  ;  but,  if  the  scale  was  7,  in  lieu  of  ten,  then,  what 
is  now  true  for  9  and  11,  would  be  then  true  for  6  and  8. 

6  We  may  change  numbers  from,  the  scale  of  ten  to 
any  other  scale,  by  dividing  the  number  by  the  number 
denoting  the  scale,  continually  saving  the  remainders 
and  forming  a  new  number  by  them. 

Example — Change  63  into  an  equivalent  number, 
wherein  the  value  of  the  digit  shall  increase  in  a  five-fold 
proportion  ;  in  other  words,  where  the  root  of  the  scale 
shall  be  5. 

5)63 

5)12(3=  first  remainder. 
5)2(2=  second  remainder. 
0(2=  third  remainder. 

Now  223  is  the  value  of  63,  in  a  scale  where  the  num- 
bers increase  in  a  five-fold  proportion.  Change  3714  to 
its  equivalent  value  on  a  scale  of  4.  Ans.  322002. 

Numbers  may  also  be  considered  as  arising  from  the 
continued  multiplication  of  certain  factors. 

Ji  perfect  number,  is  one  which  is  equal  to  the  nun  of 
all  its  divisors.  They  are  not  numerous,  and  may  be 
expressed  thus  : 

2  (22 — 1)=2X3=6. 

22(23 — 1)  =4X7=28. 
2  4(2  5—l)  =  16X  31=496. 
2  6(27— 1)  =  64X  127=8128. 
210(2U— *)  =  1024X  2047=2096128. 


A  PRACTICAL    SYSTEM 

OF 

BOOK-KEEPING, 

FOR 

MECHANICS  AND  RETAILERS. 


BOOK-KEEPING  is  the  method  of  recording  a  system- 
atic account  of  business  transactions. 

It  is  of  two  kinds — Single  and  Double  Entry.  The 
former,  only,  will  be  noticed  in  this  work.  On  account 
of  the  simplicity  of  Single  Entry,  it  is,  perhaps,  the  best 
wliich  can  be  recommended  to  farmers,  mechanics,  and 
retailers.  It  consists  of  two  principal  books — the  Day 
Book,  or  Wast*  Book,  and  the  Lcger,  and  one  auxiliary 
book,  the  Cas/trBook. 


TIIE  DAY  BOOK. 

This  book  is  ruled  with  a  column  on  the  left  hand 
for  the  date,  and  three  columns  on  the  right,  the  first, 
for  the  folio  or  page  of  the  Leger,  to  which  the  account 
is  transferred;  and  tiie  last  two  for  dollars  and  cents. 

This  book  exhibits  a  minute  history  of  business  trans- 
actions in  the  order  of  time  in  which  they  occ  ur,  with 
every  circumstance,  necessary  to  render  the  tra  asaction 
plain  and  intelligible. 

17  "  * 


CINCINNATI,  1833. 


Jan.   4 


6 


«      8 


«     11 


a     14 


14 


15 


David  Judkins,  Dr. 

To    10  Ibs.  coffee  at  17cts.$l  70 
"  25  Ibs.  sugar,  at  LO  cts.  2  50 


Timothy  W.  Coolidge,         Dr. 
To   1  bl.  sugar,  weighing 

135  Ibs.  neat,  at  8*  cts.  $11  47i 
1  bag  coffee,  98  Ibs.  at 
15  cents,  14  70 


Geo.  H.  Eaton,  Dr. 

To    1  bl.  flour,  $3  874 

«  1  Ib.  Y.  H.  Tea,  I  i2i 

"  1  keg  lard,  neat  weight 
60  Ibs.,  at  6i  cts.  375 


David  Judkins, 
By  cash  on  account, 


Cr. 


James  Wilson,  Dr. 

To  3  weeks  boarding,  at  2  dollars 
per  week, 


Hiram  Ames,  Dr. 

To    12  yards  broadcloth  at  6  dol- 
lars per  yard,  $72  00 
"  30  yds.  muslin  at  14  cts.  4  20 

Cr. 

By  an  order  on  J.  Jones,  for  Gro- 
ceries, $51  00 
"  Cash,  20  00 


Timothy  W.  Coolidge,     Cr. 
By  a  bill  of  carpenter  work, 


James  Wilson,  Dr. 

To    1  bl.  vinegar,  $3  25 

"  1  keg  lOd.  nails,  weight 
"111  Ibs.  at  8  cts.  888 


76 


71 


25 


12 


CINCINNATI,  1833. 


Jan. 19 
«  21 
«  25 


George  Ha  rail  ton, 
By  1  set  of  Fancy  chairs, 


George  H.  Eaten, 
By  cash,  10  balance  account, 


R  >bert  Young,  Dr. 

To   cash  on  account,  $3  00 

"  10  Ibs.  N.  O.  sugar,   at 
«   10  cents,  1  00 

«  12  Ibs.  coffee  at  163  cts.  2  00 


29  T( 


"    31 
Feb. 


«     13 


Cr. 


Cr. 


«  Alb.Y.H.  Tea, 


62* 


Jackson  Moore,  Dr. 

21  Ibs.  Ham,  at  10  cts.  $2  10 
«  I  box  soap,  30  Ibs.  at  5 


cents, 


150 


David  Judkins, 


Dr. 


«    30  To   12  bis.  apples  at  1 5  cents, 


By  47  bush,  corn,  at    0  cents, 


Cr. 


James  Wilson, 
By  cash  on  account, 


Cr 


Thomas  Hilton,  Dr 

3  To    cash,  $5  00 

«  1  Ib.  Y.  H.  Tea,  1  06 

«  16  Ibs.  Rice,  at  64  cts.     1  00 

Cr. 
By  13  days  labor,  at  87*  cts. 


George  Hamilton,  Dr 

10  To    1  canister  Imperial  Tea,  14 
Ibs.  at  $1  87  i  per  Ib. 


Hiram  Ames, 


Cr. 


By  order  on  E.  Disney  for  goods, 


15 


11 


•76 


iO 


CINCINNATI,  1833. 


.Feb.  15 


21 


27 


March  6 


James  Wilson,  Dr. 

To   10  gais.  molasses,  at  40 

cents,  $4  00 

"  4  Ibs.  Old  Hyson  Tea, 
at  93  cents,  372 


Robert  Young,  Dr. 

To    1  box  sperm  candles, 
"25  Ibs.,  at  30  cts.  perlb.  $7  50 
"  2  bu.  dried  fruit,  at  $1 
"  25  per  bushel,  2  50 


Robert  Young,      \  Cr. 

By  12  cords  of  wood  at       $2  25 


Ames  &,  Smith,  Dr 

To    18  Ibs.  sole  leather,  at 
"  25  cents,  $4  50 

"  1  side  upper  leather,          2  75 
"3  calf  skins,  $125,  375 


Hiram  Ames,  Dr 

To  500  feet  whits  pine  boards,  ai 
"  $12  50  per  M.  $6  25 

"  25  bu.  potatoes,  at  50  cts.  12  50 
«  1  ton  of  Hay,  10  00 


David  Judkins,  Dr 

To   200  Ibs.  flour,  at  $175 


15 


Thomas  Hilton, 


Dr 


To  cash  paid  his  order  to  William 
Cuolirlge, 


George  Hamilton,  Dr 

To  1  copy  Whelpley's  Com- 

pend,  $1  2i 

"  1  ream  letter-paper,  4  51 

«  1  doz.  Spelling  books,  1  0( 


27 


11 


18 


cts. 
00 

72 
00 

00 

75 
50 

75 


CINCINNATI,   1833. 


Mar.  20 


23 


Ames  and  Smith,          ,     Cr 
25  By  1  hhd.  sugar,  weight  1317  Ib 
neat,  at  74  cts. 


«    28 


«     30 


31 


Theodore  P.  Letton,         Dr. 
To    sharpening  his  plough,*$  1 ,00 
"  Shoeing  his  horse,  1,6 

"  Repairing  chain,  25 


.Timothy  W.  Coolidge,      Dr. 
To   2  qrs.  tuition  of  himself  at 
evening  school,  at  $3  per  qr. 


Thomas  Hilton,  Cr 

By  the  hire  of  his  horse  10  days 
at  62 i  cts.  per  day, 


T.  P.  Letton,  Dr. 

To   1  ream  wrapping  paper, 

$1,62* 

"  1  beaver  hat,  5,00 

"  1  set  silver  tea-spoons,      6,00 


Henry  C.  Sanxay,  Cr. 

By  my  order  on  him  in  favor  of 
Jno.  Torrence  for  stationary, 


Ames  and  Smith,  Dr. 

To  2000ft.  clear  pine  boards,  at 

$20  per  M.  $40,00 

«  500  common  do.  at  $8,  4,00 
«  5000  shingles,  at  $2,25  1 1,25 
"  Cash  to  balance  account,  32,52 


Jackson  Moore, 
By  painting  my  house, 


Cr. 


98 


cts. 


874 


00 


25 


774 


624 


874 


77 


00 


6                             CINCINNATI,  1833. 

March  31 

«      31 

«       31 
«       31 
«       31 

Thomas  Hilton,                 Dr. 
To   4  bu.  wheat,  at  $1  25,  $5  00 
«  1  bl.  mess  pork,                 9  00 
"  2  bu.  salt,  at  50  cts.           1  00 
"  8  Ibs.  brown  sugar,  11  cts.   88 

2 

4 
3 

2 

f- 

$ 
15 

14 
I) 
14 
11 

cts 

88 

00 
00 
00 
50 

George  Hamilton,             Dr. 
To   12  cedar  posts,  at  25c  $300 
«  1  plough,                          937* 
"  1  scythe,                           1  62d 

Jackson  Moore,                 Dr. 
To      repairing  his   wagon     and 
plough, 

Thomas  Hilton,                  Cr. 
By  1  pair  shoes,                    $1  50 
"  1  mahogany  table,          12  50 

George  Hamilton,             Cr. 
By  an  order  on  J.  Hulse,    $5  00 
«  cash,                                   650 

END  OF  THE  DAY  BOOK. 

r 

THE    LEGER. 

THIS  book  is  used  to  collect,  the  scattered  accounts  of  the  Day  Book,  and 
toarrunpe  all  tliat  relates  to  each  individual,  into  one  separate  statement. 
The  i-usiness  of  transferring  the  accounts  from  the  Day  Book  to  the  Leger, 
is  called  posting. 
The  Leger  is  ruled  with  a  double  line  in  the  middle  of  the  page,  to  sepa- 
rate the  debits  from  the  credits.     Each  side  has  two  columns  for  dollar*  and 
cents,  one  for  the  page  of  the  Day  Book,  from  which  the  particular  item  ia 
brought,  and  a  column  for  the  date. 
When  an  account  is  posted,  the  page  of  the  Leger  on  which  this  account 
is  kept,  is  written  in  the  column  for  that  purpose  in  the  Day  Book,  and  also 
the  page  of  the  Day  Book  from  which  the  account  was  posted,  is  written  in 
the  2d  column  of  the  Leger. 
Tn  posting,  begin  with  the  first  account  in  the  Day  Book,  which  you  will 
perceive  is  the  name  of  David  Judkins.    Enter  his  name  in  »he  first  page  of 
the  Leger,  in  a  large,  fair  hand,  with  Dr.  on  the  left  and  Cr.  on  the  right.— 
As  there  are  several  articles  charged  to  D.  judkins  on  the  4th  of  January, 
instead  of  specifying  each  article  in  the  Leger,  we  merely  say,  For  Sundries, 
and  enter  the  amount  in  the  proper  columns  —  see  Leger,  pane  1. 
The  Leger  has  an  index  or  alphabet,  in  which  the  narn«3  of  persons  are 
arranged  under  their  initial  letters,  with  the  page  in  the  Leger  where  the 
account  may  be  found. 

ALPHABET  TO  THE  LEGER. 

A 

Ames,  Hiram        -       2 
Auies  &  Smith    -        4 

I    J 

Judkins,  David     •        1' 

R 

B 

Balance                         5 

K 

s 

Sanxay,  H.  C.      -       4 

C 

Coolidge,  T.  W.   -        1 

L 
Letton,  T.  P.        -       4 

T 

D 

M 
Moore,  Jackson             3 

U 

E 
Eaton,  George  H-          1 

N 

V 

F 

0 

W 
Wilson,  Jas.         -       2 

G 

P 

X 

Y 

Young,  Robt.         -        3 

H 

Hilton,  Thomas    -        2 
Hamilton,  Geo.     -        3 

a 

Z 

I  1       Dr.                    David  Judkins,                    Cr. 

183& 
Jan.  4 
«  30 

Mar.7 

Apl.l 

NOTE.- 

tno  sums 
and  for  w 
that  the 
person  in 

Fo   sundries  2 
«     Apples    3 
«      Flour     4 

To     balance 
of    account 
bro't  down, 

-The  Dr.  on  the  left 
entered  on  that  side 
'liich  they  owe  you. 
sums  entered  on  that 
ider  whose  account  t 

4 
9 

3 

16 

= 

4 

Land 
ofth 
TheC 
side  c 

hey  st 

20 
)0 
30 

70 
30 

sideo 
epage 
/r.  on 
>f  tlie 
and,  a 

Jan.  6 
"  30 

r  the  page 
are  those 
he  right  1 
page  are 
id  for  wl 

By  cash    2    t 
«   Corn  3   I 
"      Bal    '   ^ 

•11 

signifies  debtor, 
for  articles  sold  t 
)and  side  signifies  c 
"or  articles  receive 
irh  you  owe  him. 

J|00 
3  40 
1  30 

3  70 

and  that 
o  others, 
redit,  or 
1  of  the 

Timothy  W.  Coolidge. 

Jan.  ^ 

Mar.  20 

Apl.  1 

NOTE.- 
idge  has 
It  appea 
which  dt 
and  tlier 

To   sundries  "2 
"     Tuition    5 

To   balance 

-The  Dr.  aide  of  this 
received  of  Die,  and 
•a  that  the  total  aino 
duct  the  $25  00  (whi 
3  will  remain  a  balan 

R 

32 

7 

aero 
the  C 
unt  o 
-  ti  ata 
ceof 

17* 
00 

17* 
174 

unt  ah 
r.  side 
f  my? 
ndsto 
S717i 

Jan  14 

Apl.  4 

nws  the  a 
shows  w 
iccountaf 
his  credit 
due  me. 

By  work   2f< 
"  Balance 

\ 

\ 

mount  of  articles  ! 
lat  F  have  receive.^ 
'ainst  liim  if  $32  1 
on  the  ri;jht  of  the 

,500 
7  17* 

g.17* 

Hr.  Cool- 
of  1  hn. 
7J,  from 
account) 

George  H.  Eaton. 

Jan.  6 

NOT*.- 
ttellcHi 
1*  fully  c 

To  sundries  ,$ 

-Thte  account  prese 
e  J.  H  Eaton  noihii 
loaed. 

1    € 

nts  e 
g<  an 

75 

iua)  si 

1  tlrat 

Jan21 

ma  on  ho 
he  owes 

By  cash  |3 

th  sides;  hence  H  1 
me  nothing.     Tlu 

875" 

s  evident 
;  sccount 

Dr. 

Hiram  Ames.                      Cr.       2 

Jan  1  i  r 
Mar.O 

Fo  sund.'  2 
"  Corn  4 

713 

28 

2U 

75 

Jan.  11 
Feb.  13 
Apl.    1 

By  sund. 
"  Order 
"       Bal 

2 
3 

7) 
10 
23 

00 
00 

95 

104 

95 

104 

95 

Apl.  1 

To  Bal 

,23 

95 

Thomas  Hilton. 

iFeb.3 
Mar  8 
"  21 

To  sund.  S 
«  Cash.  4 
«  do.  G 

7 
18 
15 

Ot> 

75 
88 

Feb.  3 
Mar23 

«    31 
Apl.  1 

By  labor, 
"  hire  of 
horse, 
«  Sund. 
«      Bal 

4 

5 
6 

11  : 

OS 
14  ( 
10( 

m 

>5 
)0 
)6i 

41 

69 

41  e 

>9 

Apl.  1 

To  Bal 

10 

06* 

James   Wilson. 

Jan.  11 

«  11 
Feb.  21 

To    boar- 
ding, 
>  "  Sunds. 
L  "     do. 

2    6 

2  12 

4     7 

00 
13 
72 

Jan.  31 
April  1 

:  By  cash 
"     Bal 

3 

15 
10 

00 

85 

25 

S5 

25 

85 

April  ] 

I    To   bal- 
ance bro't 
down, 

10 

85 

s=s 

NOTE.  —  When  an  account  is  settled  only,  and  sot  fully  paid,  as  in  the  above, 
arid  several  preceding  accounts,  the  balance,  whether  it  he  in  your  favor  or 
against  you,  is  brought  down  and  placed  distinctly  by  itself,  and  serves  for  the 
beginning  of  a  new  account,  as  you  perceive  has  been  done  in  the  above  exam- 
ple, the  balance  being  f  1085. 

3       Dr.                  George   Hamilton.                  Cr. 

Feb.  10 
Mar.15 

To    Tea, 
"   Sund. 

3  20 

25 
75 

Jan.  T 
Mar  3 

J   By  chairs  ft 
1     «  Sunds.  6 

11 

00 
50 

"     31 

"     do. 

0  14 

00 

Apl. 

L     Balance, 

10 

50 

47 

00 

4? 

00 

April  1 

To   Bal 

10 

50 

Robert  Young. 

Jan.  27 
Feb.  15 

To   sund- 
ries,   [ 

"    do.      * 

*    6< 
I  1.0  ( 

52* 
)0 

Feb.27 

By  wood, 

4 

tf 

'00 

Apl.    1 

«    Bal. 

101 

m 

27  ( 

)0 

2" 

^00 

Apl.l 

By  J?aZ. 

It 

1374 

NOTE.—  I 

n  the  above  ace 

cnmt  th 

ediffi 

jrence  bel 

ween  the  Dr. 

an 

d  C 

r.  side  is 

$10  37^,  by  which  I  perceive  that  the  balance  against  me,  in  favor  of  Robert 
Voting,  is  S  10  37*. 

Jackson  Moore, 

Jan.  29  To     Sun- 

Mar.31  J 

Jy  paint- 

dries, c 

3  e 

•0 

ing, 

a 

21 

00 

Mar.31 

«  Sunds.  t 

6C 

Apl.    1 

«      Bal. 

11  4 

0 

210 

0 

21 

00 

~  Apl.    1  1 

}y  bal- 

a 

ince  bro't 

-, 

own, 

11 

40 

Dr. 

Ames  4*  Smith.                     Cr.      4 

Mar.  6 
«     30 

To  Sundries 
"     do. 

4 
5 

1     i 

87 

00 

77 

By  sugar,  1 

fa 

= 

77 
77 

98 

77 

NV*TK.—  • 

amount  or 

This  account,  like 
i  the  Dr.  side  Mng  . 

the 
ust 

on 
equ 

3  on 
al  to 

the  first  p 
that  on  the 

age,  is  fully  c 
Cr. 

osed, 

the 

T.  P.  Lctton. 

Mar.  20 
"     20 

To  Sundries 
«      do. 

5 

5 

'2 

87^ 
ti'2i 

Ap.  1 

By  Bal. 

15 

50 

15 

50 

15 

50 

April  1 

To   balance 

15 

50 

NOTE.- 

ence  is,  th 
tail,  hi  the 

In  the  above  accoirr 
at  T.  P.  Letton  ow< 
Day  Book,  page  5. 

itt 

>a  n 

iere 
ie  $ 

is  ni 
155(1 

i  sum  on  tl 
for  sundrj 

e  Cr  side,  and 

articles  ezpret 

the! 
eedii 

ifer- 
ide- 

Henry  C.  Sanxay. 

APi  i: 

Ho  balance, 

1s 

1C 

;p| 

Mar.28 

By  my 
order,  5 

12  1 

10  f, 

371 

r     = 

Apl.    1 

By  bal 

121 

^7i 

NOTE.—  Tn  f  his  account  it  will  he  perceived  that  as  there  in  no  amount  char- 
ged to  B.C.  Sanxay,  on  the  Dr.  side,  I  owe  hint  $12  t!7|. 

5       Dr. 

Balance.                            Cr. 

April  1 

ToD.  Judkins,          11 
"  T.  W.  Coolidge,  1  1 
••    H.Ames,               2 
"  T.  Hilton,              2 
"   J.  Wilson,             2 
"  G.  Hamilton,         3 
"  T.  P.  Letton,        4 

4 

•2: 
lO 
0 
10 
15 

W 
7* 

)5  ! 

So 

}.'i 
>o 

April  1 

By  R.  Youn£, 
"   J.  Moore, 
"  II.  Sanxay, 

! 

101 

i  ) 

12 

«7J 
40 
-7| 

82 

54 

34 

G5 

NOTK.—  This  account  exhibits  the  exact  state  of  your  hooks.    It  is  made 
from  the  preceding  accounts  in  the  Leser.    The  Dr.  side  is  an  exhibit  of  the 
amounts  due  to  you  by  others,  arid  the  Cr.  side  the  amounts  due  by  you  to  oth- 
ers.   It  is  noi  strictly  necessary  that  this  account  'should  he  introduced  in  tne 
Le<jer  in  single  entry  :  it  will  he  found  convenient,  however,  to  balance  the 
book  at  stated  intervals,  and  transfer  the  balances  to  the  new  accounts  1  elow, 
1  as  in  the  preceding  Leger,  and  when  that  is  done,  a  balance  account  like  the 
"  above  will  be  found  convenient,  as  presenting,  at  one  view,  the  exact  state 
of  your  Leger. 
FORM  OF  A  BILL  FROM  THE  PRECEDING. 

» 
January    4 

30 
March       8 

January    8 
30 

Mr.  David  Judkins, 
To  Edward  Thomson,  Dr. 
To    lOlbs.  coffee  at  17  cts.    -        -        -        -            §1  7C 
25  Its.  sugar  at  10  cts.        ....          2  5( 

4 
0 

3 

1(5 

12 

20 
00 
50 

TO 
40 

12  bbls.  apples  at  75 
200  IDS.  flour  at  $1 

cts. 

.        .        .        . 

00 
40 

Cr. 

«3 

47  bushels  corn  at  20  els.    ....         9 

Errors  excepted.                                             Balance  due 
Rec'd  payment  in  full,      EDWARD  THOMSON. 

4 

30 

THE  CASH  BOOK. 

The  Cash  Book  is  used  to  record  the  daily  receipts  and  payments  of  mo- 
ney.   It  ia  ruled  nearly  the  same  as  the  Leger  ;  the  Dr.  side  exhibits  the 
amount  of  money  received,  and  the  Cr.  side,  the  amount  paid  out.  Subtract 
the  sum  of  the  Or.  from  that  of  the  Dr.  and  the  balance  will  always  be  equal 
to  the  amount  of  cash  on  hand. 

FORM  OF  A   CASH  BOOK. 

Dr. 

Cash. 

Cr. 

1833 
Jan.  1 
"    1 
•    1 
"    1 

To  cash  on  ha:.d, 
Cash  rec'd  oC  J.  Young. 
"        ''     H.  Sanxay. 
"        "    I).  Judkins 

7; 

H 

"25 
1C 

81 
40 
GO 
00 

J833 
Jan.  1 

"    1 
"    1 

By  rent  of  house  paid 
T.  P.  Letton, 
Paid  note  to  R.  Hand, 
Family  expenses, 
By  cash  on  hand, 

18 
5(J 
4 
52 

UO 

00 

37 
84 

Li>5 

21 

12o21 

Jan.  2 
"    2 
"    2 

To  cash  on  hand, 
of  T.  Coolidge, 
Cash  found  on  Main  St. 

59 

2: 
29 

81 
16 
00 

Jan.  2 

"    2 

By  cash  paid  Ames  & 
Smith, 
By  cask  an  hand, 

20 
85 

00 
00 

'Or, 

00 

105  00  ! 

Jan.  3 

To  cash  on  hand, 

65 

00 

Talbott,   J.;, 


THE  UNIVERSITY  OF  CALIFORNIA  LIBRARY 


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LHABLE  SCHOOL  BOOKS 

PUBU    1ED  BY  E.  MORGAN  & 
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:•    -v  •    \'.fJ:?,oyED  MATHEMATICAL  WGMS.       '] 

•  ;;g  works  foriK  a  complete  Elementary  COUV,-K  ol 

JTatliott's  I*iifle  Arithmetic   tie 

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p;  <  :  •,  •  B  - 


Practical  Arithmetic,  ;; 

!     '   ••    h    •  added  ^he  PRicrsf*-:.i-/ 
nn;^^  :-;;:  ,,:  Deviations,  ali  h  ^,; 
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lgetora.—  AH  Elementary  Trea 
?3,  designed  to  facilitate  the  comprehension,  demonstrs 
id  applicuuon  ofche  leading  prvacipies  of  Algebrs      v  0 
luch«l,  A.,,  M,,  Prufes&OT  o*"  Mathematics  aati'  Kstura!  fhi 
3«j)hy,  Cincinnati  s 

Iflitehel'9;- 

',  <  .  ,  ;  '  •  -  /  ,  in  w  hi  cb  rhe  reasoning 

ii  analysis,  s-uti  the  proportions 
according  to  tlieir  immediate  deper  •-'*  ; 
;,      By  O,  M~  Mitcliel,  A.  M.»  Professor  02  iV 
id  ,-jaturai  Philosophy,  Cincinnati  college. 

A  Unirersal  Kef  to   the 

,—  Fa  which  the  numencal- 

the  Appendix  of  this  Arithme'; 
!abo?  of  algebru;-'  ? 
one  foarth  the  us:  ••„.•; 


^>iHtions>  to   .• 
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<;;8sentiai  math 


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ch  Ea-tion, 


